Abstract

The anomalous-diffraction approximation of Van de Hulst is used to derive closed-form expressions for the absorption and extinction efficiencies of the long absorbing cylinder at normal and oblique incidence. The predictions of these expressions are compared with those of the exact theory for cylinders of various sizes, refractive indices, degrees of absorption, and orientations. A correction for the approximate solution at very oblique incidence is suggested.

© 1970 Optical Society of America

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References

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  1. H. C. van de Hulst, Optics of Spherical Particles (Duwaer, Amsterdam, 1946).
  2. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  3. D. Deirmendjian and R. J. Classen, J. Opt. Soc. Am. 51, 620 (1961).
    [Crossref]
  4. D. H. Napper and R. H. Ottewill, Kolloid-Z. Polymer 218, 41 (1967).
    [Crossref]
  5. F. D. Bryant, dissertation, Auburn University, 1968.
  6. F. D. Bryant and P. Latimer, J. Coll. Interf. Sci. 30, 291 (1969).
    [Crossref]
  7. J. M. Greenberg, in Stars and Steller Systems (Chicago U. P., Chicago, Ill., 1968), Vol. VII, p. 221.
  8. The extinction, scattering, and absorption efficiencies are here represented in Deirmendjian’s3 notation, as Kext, Ksca, and Kabs, respectively; in Van de Hulst’s treatment, the same quantities are called Qext, Qsca, and Qabs. They are related by Kext = Kabs + Ksca.
  9. Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun, Natl. Bur. Std. (U. S.) Handbook Appl. Math. Ser. 55 (U. S. Govt. Printing Office, Washington, D. C., 1964; Dover, New York, 1965).
  10. G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge University Press, Cambridge, 1966).
  11. A. C. Lind and J. M. Greenberg, J. Appl. Phys. 37, 3195 (1966).
    [Crossref]

1969 (1)

F. D. Bryant and P. Latimer, J. Coll. Interf. Sci. 30, 291 (1969).
[Crossref]

1967 (1)

D. H. Napper and R. H. Ottewill, Kolloid-Z. Polymer 218, 41 (1967).
[Crossref]

1966 (1)

A. C. Lind and J. M. Greenberg, J. Appl. Phys. 37, 3195 (1966).
[Crossref]

1961 (1)

Bryant, F. D.

F. D. Bryant and P. Latimer, J. Coll. Interf. Sci. 30, 291 (1969).
[Crossref]

F. D. Bryant, dissertation, Auburn University, 1968.

Classen, R. J.

Deirmendjian, D.

Greenberg, J. M.

A. C. Lind and J. M. Greenberg, J. Appl. Phys. 37, 3195 (1966).
[Crossref]

J. M. Greenberg, in Stars and Steller Systems (Chicago U. P., Chicago, Ill., 1968), Vol. VII, p. 221.

Latimer, P.

F. D. Bryant and P. Latimer, J. Coll. Interf. Sci. 30, 291 (1969).
[Crossref]

Lind, A. C.

A. C. Lind and J. M. Greenberg, J. Appl. Phys. 37, 3195 (1966).
[Crossref]

Napper, D. H.

D. H. Napper and R. H. Ottewill, Kolloid-Z. Polymer 218, 41 (1967).
[Crossref]

Ottewill, R. H.

D. H. Napper and R. H. Ottewill, Kolloid-Z. Polymer 218, 41 (1967).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Optics of Spherical Particles (Duwaer, Amsterdam, 1946).

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

Watson, G. N.

G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge University Press, Cambridge, 1966).

J. Appl. Phys. (1)

A. C. Lind and J. M. Greenberg, J. Appl. Phys. 37, 3195 (1966).
[Crossref]

J. Coll. Interf. Sci. (1)

F. D. Bryant and P. Latimer, J. Coll. Interf. Sci. 30, 291 (1969).
[Crossref]

J. Opt. Soc. Am. (1)

Kolloid-Z. Polymer (1)

D. H. Napper and R. H. Ottewill, Kolloid-Z. Polymer 218, 41 (1967).
[Crossref]

Other (7)

F. D. Bryant, dissertation, Auburn University, 1968.

H. C. van de Hulst, Optics of Spherical Particles (Duwaer, Amsterdam, 1946).

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

J. M. Greenberg, in Stars and Steller Systems (Chicago U. P., Chicago, Ill., 1968), Vol. VII, p. 221.

The extinction, scattering, and absorption efficiencies are here represented in Deirmendjian’s3 notation, as Kext, Ksca, and Kabs, respectively; in Van de Hulst’s treatment, the same quantities are called Qext, Qsca, and Qabs. They are related by Kext = Kabs + Ksca.

Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun, Natl. Bur. Std. (U. S.) Handbook Appl. Math. Ser. 55 (U. S. Govt. Printing Office, Washington, D. C., 1964; Dover, New York, 1965).

G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge University Press, Cambridge, 1966).

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Figures (4)

Fig. 1
Fig. 1

A long, circular cylinder with a normal to the axis making angle θ with the incident beam. Two different polarizations of the light are defined: case I and case II. E of case I and H of case II are in the plane defined by k ˆ and the cylinder axis. Also shown are central rays that enter the cylinder without refractive bending, as is assumed in ordinary anomalous-diffraction theory (ray 1), with refractive bending according to Snell’s law (ray 2), and with an intermediate amount of refractive bending (ray 3).

Fig. 2
Fig. 2

The extinction and absorption efficiencies Kext and Kabs of the infinite cylinder of refractive index n = 1.05 in various orientations: from exact expressions for case I polarization (– – –), and from the anomalous-diffraction approximation (——). Point codes for identifying absorption coefficients for the various curves: (○) tanβ = 0.0; (△) tanβ = 0.1; and (□) tanβ = 0.0. For this refractive index, the exact theory gives values that agree within 1% for 1.0 < ρ < 10. To avoid confusion, the Kext curves for θ = 60°, tan β = 0.1 are omitted for p > 7.

Fig. 3
Fig. 3

The extinction and absorption efficiencies for an infinite cylinder of refractive index n = 1.50 for three orientations: as calculated from exact relations for case I polarization (– – –) and for case II polarization (· · · ·), and from the anomalous-diffraction approximation (——). Point code for absorption coefficients: (○) tanβ = 0.0 (no absorption); (△) tanβ = 0.1; (□) tanβ = 0.5. For θ = 60°, only the extinction curve for tanβ = 0.0 and the absorption curve for tanβ = 0.1 are shown.

Fig. 4
Fig. 4

Kext curves for nonabsorbing cylinders (tanβ = 0) at oblique incidence (θ = 60°). Anomalous-diffraction curves were calculated without corrections for refractive bending (——), see ray 1 of Fig. 1, and with such corrections (– · – · –), see ray 3 of Fig. 1. Exact curves were calculated for case I polarization (– – –) and case II (· · · ·). The corrections are seen to bring the approximate K curves more nearly in phase with the exact ones.

Tables (1)

Tables Icon

Table I Ratio of the approximate to exact values of Kexti = K(approx)/Ki(exact), i = I or II] for small values of ρ where θ = 30° and tanβ = 0.0. ΩI and ΩII are the ratios for case I and case II polarizations, respectively.

Equations (11)

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K ext ( ρ * ) = 2 Re 0 π / 2 [ 1 - exp ( - i ρ * cos γ ) ] cos γ d γ ,
K ext ( ρ * ) = 2 Re { 0 π / 2 [ 1 - cos ( ρ * cos γ ) ] cos γ d γ + i 0 π / 2 sin ( ρ * cos γ ) cos γ d γ } .
A = ρ * 0 π / 2 sin 2 γ sin ( ρ * cos γ ) d γ ,
B = ρ * 0 π / 2 sin 2 γ cos ( ρ * cos γ ) d γ .
K ext ( ρ * ) = π Re [ H 1 ( ρ * ) + i J 1 ( ρ * ) ] .
K abs ( 4 x n ) = 0 π / 2 [ 1 - exp ( - 4 x n cos γ ) ] cos γ d γ ,
K abs ( 4 x n ) = 0 π / 2 [ 1 - cosh ( 4 x n cos γ ) ] cos γ d γ + 0 π / 2 sinh ( 4 x n cos γ ) cos γ d γ .
C = - 4 x n 0 π / 2 sin 2 γ sinh ( 4 x n cos γ ) d γ ,
D = 4 x n 0 π sin 2 γ cosh ( 4 x n cos γ ) d γ .
K abs ( 4 x n ) = - π / 2 [ L 1 ( 4 x n ) - I 1 ( 4 x n ) ] .
ρ = ρ / 2 { sec θ + sec [ sin - 1 ( sin θ / n ) ] } ,