Abstract

The generalized eikonal approximation method is applied to the study of light scattering by a dielectric medium. In this method, the propagation of light inside the medium is assumed to be rectilinear, as in the usual eikonal method, but with a parameterized propagator which is used to include the edge effect and ray optics behavior at the limit of very short wavelengths. The resulting formulas for the intensity and extinction efficiency factor are compared numerically and shown to agree excellently with the exact results for a homogeneous dielectric sphere.

© 1989 Optical Society of America

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References

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  1. C. J. Joachain and C. Quigg, "Multiple Scattering Expansions in Several Particle Dynamics," Rev. Mod. Phys. 46, 279–324 (1974) and references therein.
    [CrossRef]
  2. R. J. Glauber, "High-Energy Collision Theory," in Lectures in Theoretical Physics, W. E. Britten and L. C. Dunham Eds. (Interscience, New York, 1959); L. I. Schiff, "Approximation Method for High-Energy Potential Scattering," Phys. Rev. 103, 443–453 (1956); H. D. I. Abaranel and C. Itzykson, "Relativisitc Eikonal Expansion," Phys. Rev. Lett. 23, 53–56 (1969).
    [CrossRef]
  3. L. I. Schiff, "Approximation Method for Short Wavelength or High-Energy Scattering," Phys. Rev. 104, 1481–1485 (1956).
    [CrossRef]
  4. J. M. Perrin and P. Chiappeta, "Light Scattering by Large Particles I. A New Theoretical Description in the Eikonal Picture," Opt. Acta, 32, 907–921 (1985); J. M. Perrin and P. L. Lamy, "Light Scattering by Large Particles. II: A Vectorial Description in the Eikonal Picture," Opt. Acta 33, 1001–1022 (1986).
    [CrossRef]
  5. T. W. Chen, "Scattering of Light by a Stratified Sphere in High Energy Approximation," Appl. Opt., 26, 4155–4158 (1987).
    [CrossRef] [PubMed]
  6. S. K. Sharma, D. J. Somerford, and S. Sharma, "Investigation of Domains of Validity of Corrections to the Eikonal Approximation in Forward Light Scattering from Homogeneous Sphere," Opt. Acta 29, 1677–1682 (1982); T. W. Chen, "Eikonal Approximation Method for Small-angle Light Scattering," J. of Modern Opt, 35, 743–752 (1988).
    [CrossRef]
  7. T. W. Chen, "Generalized Eikonal Approximation," Phys. Rev. C30, 585–592 (1984).
  8. T. W. Chen "Combining the Born Approximation with the Eikonal Method," Phys. Rev. D29, 1839–1841 (1984).
  9. H. C. van de Hulst Light Scattering by Small Particles, (Dover, New York, 1957), Chapt. 17.
  10. D. S. Saxon, "Lecture on the Scattering of Light," UCLA Department of Meteorological Science Rep. No. 9 (1955).
  11. H. M. Nussenzveig and W. J. Wiscombe, "Efficiency Factors in Mie Scattering," Phys. Rev. Lett. 45, 1490–1494 (1980).
    [CrossRef]
  12. S. A. Ackerman and G. L. Stephens, "The Absorption of Solar Radiation by Cloud Droplets: An Application of Anomalous Diffraction Theory," J. of Atmos. Sci. 44, 1574–1588 (1987).
    [CrossRef]

1987 (2)

T. W. Chen, "Scattering of Light by a Stratified Sphere in High Energy Approximation," Appl. Opt., 26, 4155–4158 (1987).
[CrossRef] [PubMed]

S. A. Ackerman and G. L. Stephens, "The Absorption of Solar Radiation by Cloud Droplets: An Application of Anomalous Diffraction Theory," J. of Atmos. Sci. 44, 1574–1588 (1987).
[CrossRef]

1985 (1)

J. M. Perrin and P. Chiappeta, "Light Scattering by Large Particles I. A New Theoretical Description in the Eikonal Picture," Opt. Acta, 32, 907–921 (1985); J. M. Perrin and P. L. Lamy, "Light Scattering by Large Particles. II: A Vectorial Description in the Eikonal Picture," Opt. Acta 33, 1001–1022 (1986).
[CrossRef]

1984 (2)

T. W. Chen, "Generalized Eikonal Approximation," Phys. Rev. C30, 585–592 (1984).

T. W. Chen "Combining the Born Approximation with the Eikonal Method," Phys. Rev. D29, 1839–1841 (1984).

1982 (1)

S. K. Sharma, D. J. Somerford, and S. Sharma, "Investigation of Domains of Validity of Corrections to the Eikonal Approximation in Forward Light Scattering from Homogeneous Sphere," Opt. Acta 29, 1677–1682 (1982); T. W. Chen, "Eikonal Approximation Method for Small-angle Light Scattering," J. of Modern Opt, 35, 743–752 (1988).
[CrossRef]

1980 (1)

H. M. Nussenzveig and W. J. Wiscombe, "Efficiency Factors in Mie Scattering," Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

1974 (1)

C. J. Joachain and C. Quigg, "Multiple Scattering Expansions in Several Particle Dynamics," Rev. Mod. Phys. 46, 279–324 (1974) and references therein.
[CrossRef]

1956 (1)

L. I. Schiff, "Approximation Method for Short Wavelength or High-Energy Scattering," Phys. Rev. 104, 1481–1485 (1956).
[CrossRef]

Ackerman, S. A.

S. A. Ackerman and G. L. Stephens, "The Absorption of Solar Radiation by Cloud Droplets: An Application of Anomalous Diffraction Theory," J. of Atmos. Sci. 44, 1574–1588 (1987).
[CrossRef]

Chen, T. W.

T. W. Chen, "Scattering of Light by a Stratified Sphere in High Energy Approximation," Appl. Opt., 26, 4155–4158 (1987).
[CrossRef] [PubMed]

T. W. Chen, "Generalized Eikonal Approximation," Phys. Rev. C30, 585–592 (1984).

T. W. Chen "Combining the Born Approximation with the Eikonal Method," Phys. Rev. D29, 1839–1841 (1984).

Chiappeta, P.

J. M. Perrin and P. Chiappeta, "Light Scattering by Large Particles I. A New Theoretical Description in the Eikonal Picture," Opt. Acta, 32, 907–921 (1985); J. M. Perrin and P. L. Lamy, "Light Scattering by Large Particles. II: A Vectorial Description in the Eikonal Picture," Opt. Acta 33, 1001–1022 (1986).
[CrossRef]

Glauber, R. J.

R. J. Glauber, "High-Energy Collision Theory," in Lectures in Theoretical Physics, W. E. Britten and L. C. Dunham Eds. (Interscience, New York, 1959); L. I. Schiff, "Approximation Method for High-Energy Potential Scattering," Phys. Rev. 103, 443–453 (1956); H. D. I. Abaranel and C. Itzykson, "Relativisitc Eikonal Expansion," Phys. Rev. Lett. 23, 53–56 (1969).
[CrossRef]

Joachain, C. J.

C. J. Joachain and C. Quigg, "Multiple Scattering Expansions in Several Particle Dynamics," Rev. Mod. Phys. 46, 279–324 (1974) and references therein.
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig and W. J. Wiscombe, "Efficiency Factors in Mie Scattering," Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Perrin, J. M.

J. M. Perrin and P. Chiappeta, "Light Scattering by Large Particles I. A New Theoretical Description in the Eikonal Picture," Opt. Acta, 32, 907–921 (1985); J. M. Perrin and P. L. Lamy, "Light Scattering by Large Particles. II: A Vectorial Description in the Eikonal Picture," Opt. Acta 33, 1001–1022 (1986).
[CrossRef]

Quigg, C.

C. J. Joachain and C. Quigg, "Multiple Scattering Expansions in Several Particle Dynamics," Rev. Mod. Phys. 46, 279–324 (1974) and references therein.
[CrossRef]

Saxon, D. S.

D. S. Saxon, "Lecture on the Scattering of Light," UCLA Department of Meteorological Science Rep. No. 9 (1955).

Schiff, L. I.

L. I. Schiff, "Approximation Method for Short Wavelength or High-Energy Scattering," Phys. Rev. 104, 1481–1485 (1956).
[CrossRef]

Sharma, S.

S. K. Sharma, D. J. Somerford, and S. Sharma, "Investigation of Domains of Validity of Corrections to the Eikonal Approximation in Forward Light Scattering from Homogeneous Sphere," Opt. Acta 29, 1677–1682 (1982); T. W. Chen, "Eikonal Approximation Method for Small-angle Light Scattering," J. of Modern Opt, 35, 743–752 (1988).
[CrossRef]

Sharma, S. K.

S. K. Sharma, D. J. Somerford, and S. Sharma, "Investigation of Domains of Validity of Corrections to the Eikonal Approximation in Forward Light Scattering from Homogeneous Sphere," Opt. Acta 29, 1677–1682 (1982); T. W. Chen, "Eikonal Approximation Method for Small-angle Light Scattering," J. of Modern Opt, 35, 743–752 (1988).
[CrossRef]

Somerford, D. J.

S. K. Sharma, D. J. Somerford, and S. Sharma, "Investigation of Domains of Validity of Corrections to the Eikonal Approximation in Forward Light Scattering from Homogeneous Sphere," Opt. Acta 29, 1677–1682 (1982); T. W. Chen, "Eikonal Approximation Method for Small-angle Light Scattering," J. of Modern Opt, 35, 743–752 (1988).
[CrossRef]

Stephens, G. L.

S. A. Ackerman and G. L. Stephens, "The Absorption of Solar Radiation by Cloud Droplets: An Application of Anomalous Diffraction Theory," J. of Atmos. Sci. 44, 1574–1588 (1987).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst Light Scattering by Small Particles, (Dover, New York, 1957), Chapt. 17.

Wiscombe, W. J.

H. M. Nussenzveig and W. J. Wiscombe, "Efficiency Factors in Mie Scattering," Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Appl. Opt. (1)

J. of Atmos. Sci. (1)

S. A. Ackerman and G. L. Stephens, "The Absorption of Solar Radiation by Cloud Droplets: An Application of Anomalous Diffraction Theory," J. of Atmos. Sci. 44, 1574–1588 (1987).
[CrossRef]

Opt. Acta (2)

S. K. Sharma, D. J. Somerford, and S. Sharma, "Investigation of Domains of Validity of Corrections to the Eikonal Approximation in Forward Light Scattering from Homogeneous Sphere," Opt. Acta 29, 1677–1682 (1982); T. W. Chen, "Eikonal Approximation Method for Small-angle Light Scattering," J. of Modern Opt, 35, 743–752 (1988).
[CrossRef]

J. M. Perrin and P. Chiappeta, "Light Scattering by Large Particles I. A New Theoretical Description in the Eikonal Picture," Opt. Acta, 32, 907–921 (1985); J. M. Perrin and P. L. Lamy, "Light Scattering by Large Particles. II: A Vectorial Description in the Eikonal Picture," Opt. Acta 33, 1001–1022 (1986).
[CrossRef]

Phys. Rev. (3)

L. I. Schiff, "Approximation Method for Short Wavelength or High-Energy Scattering," Phys. Rev. 104, 1481–1485 (1956).
[CrossRef]

T. W. Chen, "Generalized Eikonal Approximation," Phys. Rev. C30, 585–592 (1984).

T. W. Chen "Combining the Born Approximation with the Eikonal Method," Phys. Rev. D29, 1839–1841 (1984).

Phys. Rev. Lett. (1)

H. M. Nussenzveig and W. J. Wiscombe, "Efficiency Factors in Mie Scattering," Phys. Rev. Lett. 45, 1490–1494 (1980).
[CrossRef]

Rev. Mod. Phys. (1)

C. J. Joachain and C. Quigg, "Multiple Scattering Expansions in Several Particle Dynamics," Rev. Mod. Phys. 46, 279–324 (1974) and references therein.
[CrossRef]

Other (3)

R. J. Glauber, "High-Energy Collision Theory," in Lectures in Theoretical Physics, W. E. Britten and L. C. Dunham Eds. (Interscience, New York, 1959); L. I. Schiff, "Approximation Method for High-Energy Potential Scattering," Phys. Rev. 103, 443–453 (1956); H. D. I. Abaranel and C. Itzykson, "Relativisitc Eikonal Expansion," Phys. Rev. Lett. 23, 53–56 (1969).
[CrossRef]

H. C. van de Hulst Light Scattering by Small Particles, (Dover, New York, 1957), Chapt. 17.

D. S. Saxon, "Lecture on the Scattering of Light," UCLA Department of Meteorological Science Rep. No. 9 (1955).

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram for the scattering of particle or light by a potential or dielectric medium of a sphere of radius a in the eikonal approximation.

Fig. 2
Fig. 2

Qext and Pext vs the size parameter x for light scattering by a water droplet with fixed n. The results from both the HEA and GEA methods are compared with exact results from the Mie method. The HEA modified is the HEA with (n2 − 1) replaced by 2(n − 1). Notice that the GEA results overlap with the solid curves of the exact values when x is ∼50 or higher.

Fig. 3
Fig. 3

Qext and Pext vs the imaginary part of the refractive index of a water droplet, with the real part and x fixed.

Fig. 4
Fig. 4

Qext and Pext vs the real part of the refractive index, with the imaginary part and x fixed.

Fig. 5
Fig. 5

Qext and Pext vs size parameter x for a dielectric sphere, with large refractive index.

Fig. 6
Fig. 6

Qext and Pext vs size parameter x for a dielectric sphere, with small n but large absorption.

Fig. 7
Fig. 7

Qext and Pext vs the real part of n for a dielectric sphere, without absorption at small x.

Fig. 8
Fig. 8

Relative intensity (R. I.) vs scattering angle for unpolarized light scattered by a water droplet. The upper curves are the R. I. at x = 30 multiplied by a factor of 100 and the lower curves are the R. I. at x = 10.

Equations (35)

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( 2 + k 2 ) ψ ( r ) = V ( r ) ψ ( r ) ,
ψ ( r ) r a = ψ o ( r ) + exp [ ikr ] r f ( k , k ) .
f ( k , k ) = 1 4 π d 3 rV ( r ) exp [ i q r i χ ( r ) ] ,
χ ( r ) = 1 2 k z V ( x , y , z ) d z ,
G ( r ) = exp ( ikr ) 4 π r ,
G + ( r ) = 1 2 ki δ ( x ) δ ( y ) ϴ ( z ) exp ( i Δ z ) ,
G ( r ) = 1 2 α i δ ( x ) δ ( y ) ϴ ( z ) exp ( i β z ) .
f ( k , k ) = f B 1 8 π α i d 3 r exp [ i q r i κ z i χ α ( r ) ] V ( r ) z d z V ( r ) exp [ i κ z + i χ α ( r ) ] ,
χ α ( r ) = 1 2 α z V ( r ) d z , f B = 1 4 π exp ( i q r ) V ( r ) d 3 r .
E r a = E o ( r ) + exp ( ikr ) ikr [ ( r ̂ × S ) × r ̂ ] ,
S = k 4 π i exp ( i k r ) V ( r ) E ( r ) d 3 r ,
E ( r ) = ê o E o exp ( i k r i χ ) ,
χ = 1 2 k z V ( x , y , z ) d z .
S = ê E o S ( q ) , S ( q ) = ki 4 π exp ( i q r ) V ( r ) exp ( i χ ) d 3 r ,
E ( r ) = E o ( r ) + E o ( r ) 2 α i exp [ i ( β z χ α ( r ) ] z V ( r ) exp { i [ k r β z + χ α ( r ) ] } d z ,
χ α ( r ) = 1 2 α z V ( r ) d z .
S ( θ ) = ê o S ( θ ) ,
S ( θ ) = ik 4 π S B + k 8 π α d 3 r exp [ i q r i Λ ( r ) ] V ( r ) z d z V ( r ) exp ( i Λ ( r ) ] ,
S B = d 3 r exp ( iq r ) V ( r ) .
I ( θ ) = [ 1 ( k ̂ ê ) 2 ] | S ( θ ) | 2 .
C ext = 4 π k 2 Re { S ( 0 ° ) } ,
Q ext = 4 x 2 Re { S ( 0 ° ) } ,
P ext = 4 x 2 Im { S ( 0 ° ) } .
Q ext + i P ext = 4 x 2 S ( 0 ° ) .
χ α ( b , z ) = V o 2 α [ z + Z o ( b ) ] ϴ ( Z o ( b ) | z | ) ,
S ( θ ) = ik 4 π ( 1 γ ) S B α k γ 2 0 a bdb × J o ( qb ) { exp [ i ρ Z o ( b ) / a ] 1 } ,
ρ = a V o / α γ , q = 2 k sin ( θ / 2 ) , γ = 1 / [ 1 2 ( Δ β ) α / V o ] , S B = 4 π V o 0 a b z o ( b ) J o ( qb ) db ,
S ( 0 ° ) = i 1 3 α 0 γ x 2 ( 1 γ ) ρ + 1 2 α 0 γ 2 x 2 [ 1 + 2 i ρ e i ρ + 2 ρ 2 ( 1 e i ρ ) ] ,
Q ext + i P ext = 2 { i 2 3 α 0 γ ( 1 γ ) ρ + α o γ 2 [ 1 + 2 i ρ e i ρ + 2 ρ 2 ( 1 e i ρ ) ] } .
α 0 γ = ( n + 1 ) / 2 .
Q ext + i P ext 2.0 a 1 x 2 / 3 a 2 x 4 / 3 ,
Q ext + i P ext 2 [ i 2 3 α o γ ( 1 γ ) ρ + α 0 γ 2 ] .
i 2 3 α 0 γ ( 1 γ ) ρ + α 0 γ 2 = 1 + 1 2 ( a 1 x 2 / 3 a 2 x 4 / 3 ) .
γ = ( n + 1 ) / ( 2 α o ) ,
α o = 1 + ( n 1 ) / 2 i 3 8 { 1 x 2 ρ [ a 1 x 2 / 3 a 2 x 4 / 3 ] } ,

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