Abstract

To show how apertures affect measurements of the circularly polarized components of light scattered to a detector, we develop two methods of averaging the V and I Stokes parameters over a circular aperture that collects light scattered from an optically active sphere. One method uses a two-dimensional numerical integration that is appropriate for small apertures, and the other gives analytical expressions for scattering into a solid angle of any size. We identify the aperture locations that, independent of aperture size, give an average V (and an effective degree of circular polarization) of zero for scattering from an optically inactive sphere and of nonzero for scattering from an optically active sphere.

© 1998 Optical Society of America

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References

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  1. A. Vitkin, “Polarized light and the asymmetry of life,” Opt. Photon. News 7(7), 30–33 (1996).
    [CrossRef]
  2. A. Lakhtakia, ed., Selected Papers on Natural Optical Activity (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1990).
  3. D. L. Rosen, J. D. Pendleton, “Detection of biological particles by the use of circular dichroism measurements improved by scattering theory,” Appl. Opt. 34, 5875–5884 (1995).
    [CrossRef] [PubMed]
  4. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), pp. 45–53, 62–65, 94–95, 100–101, 114, 185–193.
  5. C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
    [CrossRef]
  6. D. J. Gordon, “Mie scattering by optically active particles,” Biochemistry 11, 413–420 (1972).
    [CrossRef] [PubMed]
  7. C. F. Bohren, “Scattering of electromagnetic waves by an optically active spherical shell,” J. Chem. Phys. 62, 1566–1571 (1975).
    [CrossRef]
  8. C. F. Bohren, “Scattering of an optically active cylinder,” J. Colloid Interface Sci. 66, 105–109 (1978).
    [CrossRef]
  9. M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
    [CrossRef]
  10. M. F. R. Cooray, I. R. Ciric, “Wave scattering by a chiral spheroid,” J. Opt. Soc. Am. A 10, 1197–1203 (1993).
    [CrossRef]
  11. W. H. Pierce, “Numerical integration over the planar annulus,” J. Soc. Ind. Appl. Math. 5, 66–73 (1957).
    [CrossRef]
  12. P. Chylek, “Mie scattering into the backward hemisphere,” J. Opt. Soc. Am. 63, 1467–1471 (1973).
    [CrossRef]
  13. W. J. Wiscombe, P. Chylek, “Mie scattering between any two angles,” J. Opt. Soc. Am. 67, 572–573 (1977).
    [CrossRef]
  14. W. P. Chu, D. M. Robinson, “Scattering from a moving spherical particle by two crossed coherent plane waves,” Appl. Opt. 16, 619–626 (1977).
    [CrossRef] [PubMed]
  15. J. D. Pendleton, “Mie scattering into apertures,” J. Opt. Soc. Am. 72, 1029–1033 (1982).
    [CrossRef]
  16. J. D. Pendleton, “A generalized Mie theory solution and its application to particle sizing interferometry,” Ph.D. dissertation (University of Tennessee, Knoxville, Tenn.1982), p. 90.
  17. J. Y. Son, W. M. Farmer, T. V. Giel, “New optical geometry for the particle sizing interferometer,” Appl. Opt. 25, 4332–4337 (1986).
    [CrossRef] [PubMed]
  18. J. Y. Son, “Multiple methods for obtaining particle size distribution with a particle sizing interferometer,” Ph.D. dissertation (University of Tennessee, Knoxville, Tenn., 1985).
  19. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 746.
  20. J. D. Pendleton, S. C. Hill, “Collection of emission from an oscillating dipole inside a sphere: analytical integration over a circular aperture,” Appl. Opt. 36, 8729–8737 (1997).
    [CrossRef]
  21. M. E. Rose, Elementary Theory of Angular Momentum (Dover, New York, 1995), p. 50.
  22. G. Arfken, Mathematical Methods for Physicists, 3rd. ed. (Academic, San Diego, Calif., 1985), pp. 198–200, 253, 678.
  23. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 34, 124.
  24. A. N. Lowan, N. Davids, A. Levenson, “Table of the zeros of the Legendre polynomials of order 1–16 and the weight coefficients for Gauss’ mechanical quadrature formula,” Bull. Am. Math. Soc. 48, 739–743 (1942); errata 49, 939 (1943).
    [CrossRef]
  25. S. L. Belousov, Tables of Normalized Associated Legendre Polynomials (Pergamon, New York, 1962).
  26. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 401.
  27. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).
  28. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), pp. 1005, 1008.

1997 (1)

1996 (1)

A. Vitkin, “Polarized light and the asymmetry of life,” Opt. Photon. News 7(7), 30–33 (1996).
[CrossRef]

1995 (1)

1993 (1)

1991 (1)

M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

1986 (1)

1982 (1)

1978 (1)

C. F. Bohren, “Scattering of an optically active cylinder,” J. Colloid Interface Sci. 66, 105–109 (1978).
[CrossRef]

1977 (2)

1975 (1)

C. F. Bohren, “Scattering of electromagnetic waves by an optically active spherical shell,” J. Chem. Phys. 62, 1566–1571 (1975).
[CrossRef]

1974 (1)

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

1973 (1)

1972 (1)

D. J. Gordon, “Mie scattering by optically active particles,” Biochemistry 11, 413–420 (1972).
[CrossRef] [PubMed]

1957 (1)

W. H. Pierce, “Numerical integration over the planar annulus,” J. Soc. Ind. Appl. Math. 5, 66–73 (1957).
[CrossRef]

1942 (1)

A. N. Lowan, N. Davids, A. Levenson, “Table of the zeros of the Legendre polynomials of order 1–16 and the weight coefficients for Gauss’ mechanical quadrature formula,” Bull. Am. Math. Soc. 48, 739–743 (1942); errata 49, 939 (1943).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd. ed. (Academic, San Diego, Calif., 1985), pp. 198–200, 253, 678.

Belousov, S. L.

S. L. Belousov, Tables of Normalized Associated Legendre Polynomials (Pergamon, New York, 1962).

Bohren, C. F.

C. F. Bohren, “Scattering of an optically active cylinder,” J. Colloid Interface Sci. 66, 105–109 (1978).
[CrossRef]

C. F. Bohren, “Scattering of electromagnetic waves by an optically active spherical shell,” J. Chem. Phys. 62, 1566–1571 (1975).
[CrossRef]

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), pp. 45–53, 62–65, 94–95, 100–101, 114, 185–193.

Chu, W. P.

Chylek, P.

Ciric, I. R.

Cooray, M. F. R.

Davids, N.

A. N. Lowan, N. Davids, A. Levenson, “Table of the zeros of the Legendre polynomials of order 1–16 and the weight coefficients for Gauss’ mechanical quadrature formula,” Bull. Am. Math. Soc. 48, 739–743 (1942); errata 49, 939 (1943).
[CrossRef]

Farmer, W. M.

Giel, T. V.

Gordon, D. J.

D. J. Gordon, “Mie scattering by optically active particles,” Biochemistry 11, 413–420 (1972).
[CrossRef] [PubMed]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), pp. 1005, 1008.

Hill, S. C.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), pp. 45–53, 62–65, 94–95, 100–101, 114, 185–193.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 746.

Kluskens, M. S.

M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

Levenson, A.

A. N. Lowan, N. Davids, A. Levenson, “Table of the zeros of the Legendre polynomials of order 1–16 and the weight coefficients for Gauss’ mechanical quadrature formula,” Bull. Am. Math. Soc. 48, 739–743 (1942); errata 49, 939 (1943).
[CrossRef]

Lowan, A. N.

A. N. Lowan, N. Davids, A. Levenson, “Table of the zeros of the Legendre polynomials of order 1–16 and the weight coefficients for Gauss’ mechanical quadrature formula,” Bull. Am. Math. Soc. 48, 739–743 (1942); errata 49, 939 (1943).
[CrossRef]

Newman, E. H.

M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

Pendleton, J. D.

Pierce, W. H.

W. H. Pierce, “Numerical integration over the planar annulus,” J. Soc. Ind. Appl. Math. 5, 66–73 (1957).
[CrossRef]

Robinson, D. M.

Rose, M. E.

M. E. Rose, Elementary Theory of Angular Momentum (Dover, New York, 1995), p. 50.

Rosen, D. L.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), pp. 1005, 1008.

Son, J. Y.

J. Y. Son, W. M. Farmer, T. V. Giel, “New optical geometry for the particle sizing interferometer,” Appl. Opt. 25, 4332–4337 (1986).
[CrossRef] [PubMed]

J. Y. Son, “Multiple methods for obtaining particle size distribution with a particle sizing interferometer,” Ph.D. dissertation (University of Tennessee, Knoxville, Tenn., 1985).

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 401.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 34, 124.

Vitkin, A.

A. Vitkin, “Polarized light and the asymmetry of life,” Opt. Photon. News 7(7), 30–33 (1996).
[CrossRef]

Wiscombe, W. J.

Appl. Opt. (4)

Biochemistry (1)

D. J. Gordon, “Mie scattering by optically active particles,” Biochemistry 11, 413–420 (1972).
[CrossRef] [PubMed]

Bull. Am. Math. Soc. (1)

A. N. Lowan, N. Davids, A. Levenson, “Table of the zeros of the Legendre polynomials of order 1–16 and the weight coefficients for Gauss’ mechanical quadrature formula,” Bull. Am. Math. Soc. 48, 739–743 (1942); errata 49, 939 (1943).
[CrossRef]

Chem. Phys. Lett. (1)

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

M. S. Kluskens, E. H. Newman, “Scattering by a multilayer chiral cylinder,” IEEE Trans. Antennas Propag. 39, 91–96 (1991).
[CrossRef]

J. Chem. Phys. (1)

C. F. Bohren, “Scattering of electromagnetic waves by an optically active spherical shell,” J. Chem. Phys. 62, 1566–1571 (1975).
[CrossRef]

J. Colloid Interface Sci. (1)

C. F. Bohren, “Scattering of an optically active cylinder,” J. Colloid Interface Sci. 66, 105–109 (1978).
[CrossRef]

J. Opt. Soc. Am. (3)

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

J. Soc. Ind. Appl. Math. (1)

W. H. Pierce, “Numerical integration over the planar annulus,” J. Soc. Ind. Appl. Math. 5, 66–73 (1957).
[CrossRef]

Opt. Photon. News (1)

A. Vitkin, “Polarized light and the asymmetry of life,” Opt. Photon. News 7(7), 30–33 (1996).
[CrossRef]

Other (12)

A. Lakhtakia, ed., Selected Papers on Natural Optical Activity (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1990).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-Interscience, New York, 1983), pp. 45–53, 62–65, 94–95, 100–101, 114, 185–193.

J. D. Pendleton, “A generalized Mie theory solution and its application to particle sizing interferometry,” Ph.D. dissertation (University of Tennessee, Knoxville, Tenn.1982), p. 90.

M. E. Rose, Elementary Theory of Angular Momentum (Dover, New York, 1995), p. 50.

G. Arfken, Mathematical Methods for Physicists, 3rd. ed. (Academic, San Diego, Calif., 1985), pp. 198–200, 253, 678.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 34, 124.

J. Y. Son, “Multiple methods for obtaining particle size distribution with a particle sizing interferometer,” Ph.D. dissertation (University of Tennessee, Knoxville, Tenn., 1985).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 746.

S. L. Belousov, Tables of Normalized Associated Legendre Polynomials (Pergamon, New York, 1962).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), p. 401.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), pp. 1005, 1008.

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Figures (1)

Fig. 1
Fig. 1

Position and size of the circular-aperture solid angle.

Tables (4)

Tables Icon

Table 1 Computed Values of an, bn, and cn

Tables Icon

Table 2 P, 〈V〉, and 〈I〉 for Scattering into an Aperture with α = β = 0, θmin = 5 deg, and θmax = 60 Deg, Computed with the Numerical Integration Method

Tables Icon

Table 3 P, 〈V〉, and 〈I〉 for Scattering into an Aperture with α = β = 0, θmin = 0, and θmax As Indicated, Computed with the Analytical Method

Tables Icon

Table 4 P, 〈V〉, and 〈I〉 for Scattering into an Aperture with α = 30 deg, β = 60 deg, θmin = 0, deg, and θmax As Indicated, Computed with the Analytical Method

Equations (76)

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V k 2 r 2 I R - I L I inc ,
I k 2 r 2 I R + I L I inc ,
f 1 Δ Ω Δ Ω d Ω f ,
P V / I .
x A y A z A = R α β γ x y z ,
R α β γ = R z γ R y β R z α ,
R z α cos   α sin   0 0 - sin   α cos   α 0 0 0 1 ,
R y β cos   β 0 - sin   β 0 1 0 sin   β 0 cos   β .
V = k 2 r 2 Im 2 E θ E ϕ * E inc 2 ,
I = k 2 r 2 | E θ | 2 + | E ϕ | 2 E inc 2 .
E = E inc n = 1   i n 2 n + 1 n n + 1 ia n N e 1 n 3 + - id n N o 1 n 3 + c n M e 1 n 3 + - b n M o 1 n 3 .
E θ - E ϕ = E inc exp   ikr - ikr S 2 S 3 S 4 S 1 cos   ϕ sin   ϕ ,
S 1 = n = 1 2 n + 1 n n + 1 a n π n + b n τ n ,
S 2 = n = 1 2 n + 1 n n + 1 a n τ n + b n π n ,
S 3 = n = 1 2 n + 1 n n + 1 c n π n - d n τ n ,
S 4 = n = 1 2 n + 1 n n + 1 d n π n - c n τ n ,
V = S 41 + S 42   cos 2 ϕ + S 43   sin 2 ϕ ,
S 41 Im S 1 S 3 * - S 2 S 4 * ,
S 42 Im - S 1 S 3 * - S 2 S 4 * ,
S 43 Im S 1 S 2 * - S 3 S 4 * .
I = S 11 + S 12   cos 2 ϕ + S 13   sin 2 ϕ ,
S 11 ( 1 / 2 ) | S 1 | 2 + | S 2 | 2 + | S 3 | 2 + | S 4 | 2 ,
S 12 ( 1 / 2 ) - | S 1 | 2 + | S 2 | 2 + | S 3 | 2 + | S 4 | 2 ,
S 13 Re S 1 S 4 * + S 2 S 3 * .
v 1 v 2 v 3 sin   θ   cos   ϕ sin   θ   sin   ϕ cos   θ = R α β γ - 1 sin   θ A   cos   ϕ A sin   θ A   sin   ϕ A cos   θ A
f 1 Δ Ω z A 2     d AF ,
  d AF = A   i j   w ij F ij ,
f = π Δ Ω tan 2   θ A max - tan 2   θ A min i j   w ij cos 3   θ A i f ij .
V = k 2 r 2 - i r ˆ · E × E * / E inc 2 ,
I = k 2 r 2 E · E * / E inc 2 .
M ¯ nm q k r × r z n q ρ exp im ϕ P ¯ nm cos   θ
= iz n q ρ exp im ϕ θ ˆ π ¯ nm cos   θ + i ϕ ˆ τ ¯ nm cos   θ ,
N ¯ nm q k r 1 k   × M ¯ nm q
= exp im ϕ r ˆ n n + 1 z n q ρ ρ   P ¯ nm cos   θ + d / d ρ ρ z n q ρ ρ θ ˆ τ ¯ nm cos   θ + i ϕ ˆ π ¯ nm cos   θ ,
π ¯ nm cos   θ = m P ¯ nm cos   θ / sin   θ ,
τ ¯ nm cos   θ = d / d θ P ¯ nm cos   θ ,
M σ 1 n 3 = m = - n n   f σ nm M ¯ nm 3 A ,
N σ 1 n 3 = m = - n n   f σ nm N ¯ nm 3 A ,
f onm f enm 2 2 n + 1 - 1 m i exp im γ cos   α sin   α - sin   α cos   α π ¯ nm cos   β i τ ¯ nm cos   β .
E = E inc n = 1 m = - n n   i n 2 n + 1 n n + 1 ψ nm a N ¯ nm 3 A + ψ nm b M ¯ nm 3 A ,
ψ nm a i a n f enm - d n f onm ,
ψ nm b = c n f enm - b n f onm .
M ¯ nm 3 A N ¯ nm 3 A = - i n exp ikr kr exp im ϕ A θ ˆ A π ¯ nm cos   θ A τ ¯ nm cos   θ A + i ϕ ˆ A τ ¯ nm cos   θ A π ¯ nm cos   θ A .
V I = 2 π Δ Ω n = 1 n = 1 2 n + 1 2 n + 1 n n + 1 n n + 1 m = - min n , n min n , n × - ψ nn m cross ψ nn m dot I ¯ nn m dot + - ψ nn m dot ψ nn m cross I ¯ nn m cross ,
ψ nn m dot ψ nm a ψ n m a * + ψ nm b ψ n m b * ,
ψ nn m cross ψ nm a ψ n m b * + ψ nm b ψ n m a * ,
I ¯ nn m dot μ max μ min d μ π ¯ nm μ π ¯ n m μ + τ ¯ nm μ τ ¯ n m μ ,
I ¯ nn m cross μ max μ min d μ π ¯ nm μ τ ¯ n m μ + τ ¯ nm μ π ¯ n m μ ,
V I = π Δ Ω n = 1 n = 1 m = 0 min n , n N m 2 n + 1 2 n + 1 n n + 1 n n + 1 × - Ψ nn m + cross Ψ nn m + dot I ¯ nn m dot + - Ψ nn m - dot Ψ nn m - cross I ¯ nn m cross ,
N m 1 ,   if   m = 0 , N m 2 ,   if   m 0 ,
Ψ nn m ± dot ψ nn m dot ± ψ nn , - m dot ,
Ψ nn m ± cross ψ nn m cross ± ψ nn , - m cross ,
V Δ Ω = 4 π = n = 1 2 n + 1 Im a n c n * + d n b n * ,
I Δ Ω = 4 π = 1 / 2 n = 1 2 n + 1 | a n | 2 + | b n | 2 + | c n | 2 + | d n | 2 .
Q scattering = 4 X 2 I Δ Ω = 4 π ,
Q scattering = 2 X 2 n = 1 2 n + 1 | a n | 2 + | b n | 2 + | c n | 2 + | d n | 2 .
X nM = m = - n n   D M , m n * α β γ X nm A ,
D M , m n * α β γ = exp iM α d M , m n β exp im γ .
M ¯ n , ± 1 3 = m = - n n   D ± 1 , m n * α β γ - 1 m + 1 M ¯ nm 3 A ,
D ± 1 , m n * α β γ = exp ± i α d ± 1 , m m β exp im γ ,
d ± 1 , m n β = 2 n n + 1 2 n + 1 1 / 2 × π ¯ nm cos   β ± τ ¯ nm cos   β .
M omn 3 = M ¯ nm 3 - - 1 m M ¯ n , - m 3 3 / 2 iA nm ,
M emn 3 = M ¯ nm 3 + - 1 m M ¯ n , - m 3 / 2 A nm
M σ 1 n 3 = m = - n n   f σ nm M ¯ nm 3 A ,
f onm - 1 m + 1 D 1 , m n * α β γ + D - 1 , m n * α β γ / 2 iA n , 1 .
f enm - 1 m + 1 D 1 , m n * α β γ - D - 1 , m n * α β γ / 2 A n , 1 .
V Stokes Δ Ω = 4 π = n = 1 2 n + 1 2 4 n n + 1 - 1 m = 0 n   N m Ψ nnm + cross ,
I Stokes Δ Ω = 4 π = n = 1 2 n + 1 2 4 n n + 1 m = 0 n   N m Ψ nnm + dot .
π ¯ n , 1 1 = τ ¯ n , 1 1 = 1 2 2 n + 1 n n + 1 2 1 / 2 ,
m = - n n π ¯ nm cos   β 2 = m = - n n τ ¯ nm cos   β 2 = 1 / 4 n n + 1 2 n + 1 ,
m = - n n   π ¯ nm cos   β τ ¯ nm cos   β = 0 .
m = 0 n   N m π ¯ nm cos   β 2 = m = 0 n   N m τ ¯ nm cos   β 2 = 1 / 4 n n + 1 2 n + 1 ,
m = 0 n   N m | f enm | 2 = m = 0 n   N m | f onm | 2 = n n + 1 / 2 n + 1 ,
m = 0 n   N m Re f enm f onm * = 0 .
- 1 m = 0 n   N m Ψ nnm + cross = 4 n n + 1 2 n + 1 Im a n c n * + d n b n * ,
m = 0 n   N m Ψ nnm + dot = 1 / 2 4 n n + 1 2 n + 1 | a n | 2 + | b n | 2 + | c n | 2 + | d n | 2 ,

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