Abstract

Using a plane-polarized laser of wavelength λ=543.5 or 441.6nm and spider silk, we investigated the diffraction of a transparent cylinder of diameter Dλλ5 at normal incidence. The measured pattern corresponded well to the one calculated by a rigorous solution of the theory for the problem. The birefringent index and D of the sample could be determined simultaneously. The experimental data of the scattering cross section for D<λ4 suggested that the data approached dipole radiation.

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References

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  1. M. Born, E. Wolf, Principles of Optics7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8.
    [CrossRef]
  2. A. Sommerfeld, Optics, transl. by O. Laporte, and P. A. Moldauer (Academic, New York, 1954), Sec. 34.
  3. Ref. [2], Sec. 39.
  4. Ref. [1], Chap. 11.
  5. Ref. [2], Sec. 38.
  6. H. C. van de Hulst, Light Scattering by Small Particles, (Dover, New York, 1981), Chap. 15.
  7. J. L. Lundberg, “Light scattering from large fibers at normal incidence,” J. Colloid Interface Sci. 29, 565–583 (1969).
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  8. T. Okoshi, K. Hotate, “Refractive-index profile of an optical fiber: its measurement by the scattering-pattern method,” Appl. Opt. 15, 2756–2764 (1976).
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  11. W. A. Farone, M. Kerker, “Light scattering from long submicron glass cylinders at normal incidence,” J. Opt. Soc. Am. 56, 481–487 (1966).
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  12. E. Matijevic, R. H. Ottewill, M. Kerker, “Light scattering by infinite cylinders. Spider fibers,” J. Opt. Soc. Am. 51, 115–116 (1961).
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  13. Y. Nishiyama, S. Kurita, I. Yamamoto, Y. Ishizuka, T. Watanabe, D. Kobayashi, K. Odanaka, Y. Hyogo, “Diameter and refractive index of a cylindrical thread determined by scattered light pattern,” Opt. Rev. 8, 90–94 (2001).
    [CrossRef]
  14. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 6.
  15. Y. Nishiyama, A. Sugita, M. Endo, H. Kogoe, K. Yorimasa, T. Sakai, T. Kurosawa, “Birefringence of a transparent cylinder determined by light scattering. Spider silk,” J. Fac. Edu. Hum. Sci. Yokohama Nat. Univ. 4, 11–17 (2002).
  16. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 9.
  17. L. Kirkup, “A guide to GUM,” Eur. J. Phys. 23, 483–487 (2002).
    [CrossRef]
  18. BIPM, IEC, IFCC, ISO, IUPAP and OIML, Guide to the Expression of Uncertainty in Measurement, 1st ed. (International Standards Organization, Geneva, 1993).

2002 (2)

Y. Nishiyama, A. Sugita, M. Endo, H. Kogoe, K. Yorimasa, T. Sakai, T. Kurosawa, “Birefringence of a transparent cylinder determined by light scattering. Spider silk,” J. Fac. Edu. Hum. Sci. Yokohama Nat. Univ. 4, 11–17 (2002).

L. Kirkup, “A guide to GUM,” Eur. J. Phys. 23, 483–487 (2002).
[CrossRef]

2001 (1)

Y. Nishiyama, S. Kurita, I. Yamamoto, Y. Ishizuka, T. Watanabe, D. Kobayashi, K. Odanaka, Y. Hyogo, “Diameter and refractive index of a cylindrical thread determined by scattered light pattern,” Opt. Rev. 8, 90–94 (2001).
[CrossRef]

1995 (1)

1976 (1)

1974 (1)

1969 (1)

J. L. Lundberg, “Light scattering from large fibers at normal incidence,” J. Colloid Interface Sci. 29, 565–583 (1969).
[CrossRef]

1966 (1)

1961 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 9.

Born, M.

M. Born, E. Wolf, Principles of Optics7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8.
[CrossRef]

Dandliker, R.

Endo, M.

Y. Nishiyama, A. Sugita, M. Endo, H. Kogoe, K. Yorimasa, T. Sakai, T. Kurosawa, “Birefringence of a transparent cylinder determined by light scattering. Spider silk,” J. Fac. Edu. Hum. Sci. Yokohama Nat. Univ. 4, 11–17 (2002).

Farone, W. A.

Hotate, K.

Hyogo, Y.

Y. Nishiyama, S. Kurita, I. Yamamoto, Y. Ishizuka, T. Watanabe, D. Kobayashi, K. Odanaka, Y. Hyogo, “Diameter and refractive index of a cylindrical thread determined by scattered light pattern,” Opt. Rev. 8, 90–94 (2001).
[CrossRef]

Ishizuka, Y.

Y. Nishiyama, S. Kurita, I. Yamamoto, Y. Ishizuka, T. Watanabe, D. Kobayashi, K. Odanaka, Y. Hyogo, “Diameter and refractive index of a cylindrical thread determined by scattered light pattern,” Opt. Rev. 8, 90–94 (2001).
[CrossRef]

Kerker, M.

Kirkup, L.

L. Kirkup, “A guide to GUM,” Eur. J. Phys. 23, 483–487 (2002).
[CrossRef]

Kobayashi, D.

Y. Nishiyama, S. Kurita, I. Yamamoto, Y. Ishizuka, T. Watanabe, D. Kobayashi, K. Odanaka, Y. Hyogo, “Diameter and refractive index of a cylindrical thread determined by scattered light pattern,” Opt. Rev. 8, 90–94 (2001).
[CrossRef]

Kogoe, H.

Y. Nishiyama, A. Sugita, M. Endo, H. Kogoe, K. Yorimasa, T. Sakai, T. Kurosawa, “Birefringence of a transparent cylinder determined by light scattering. Spider silk,” J. Fac. Edu. Hum. Sci. Yokohama Nat. Univ. 4, 11–17 (2002).

Kurita, S.

Y. Nishiyama, S. Kurita, I. Yamamoto, Y. Ishizuka, T. Watanabe, D. Kobayashi, K. Odanaka, Y. Hyogo, “Diameter and refractive index of a cylindrical thread determined by scattered light pattern,” Opt. Rev. 8, 90–94 (2001).
[CrossRef]

Kurosawa, T.

Y. Nishiyama, A. Sugita, M. Endo, H. Kogoe, K. Yorimasa, T. Sakai, T. Kurosawa, “Birefringence of a transparent cylinder determined by light scattering. Spider silk,” J. Fac. Edu. Hum. Sci. Yokohama Nat. Univ. 4, 11–17 (2002).

Lundberg, J. L.

J. L. Lundberg, “Light scattering from large fibers at normal incidence,” J. Colloid Interface Sci. 29, 565–583 (1969).
[CrossRef]

Matijevic, E.

Nishiyama, Y.

Y. Nishiyama, A. Sugita, M. Endo, H. Kogoe, K. Yorimasa, T. Sakai, T. Kurosawa, “Birefringence of a transparent cylinder determined by light scattering. Spider silk,” J. Fac. Edu. Hum. Sci. Yokohama Nat. Univ. 4, 11–17 (2002).

Y. Nishiyama, S. Kurita, I. Yamamoto, Y. Ishizuka, T. Watanabe, D. Kobayashi, K. Odanaka, Y. Hyogo, “Diameter and refractive index of a cylindrical thread determined by scattered light pattern,” Opt. Rev. 8, 90–94 (2001).
[CrossRef]

Odanaka, K.

Y. Nishiyama, S. Kurita, I. Yamamoto, Y. Ishizuka, T. Watanabe, D. Kobayashi, K. Odanaka, Y. Hyogo, “Diameter and refractive index of a cylindrical thread determined by scattered light pattern,” Opt. Rev. 8, 90–94 (2001).
[CrossRef]

Okoshi, T.

Ottewill, R. H.

Sakai, T.

Y. Nishiyama, A. Sugita, M. Endo, H. Kogoe, K. Yorimasa, T. Sakai, T. Kurosawa, “Birefringence of a transparent cylinder determined by light scattering. Spider silk,” J. Fac. Edu. Hum. Sci. Yokohama Nat. Univ. 4, 11–17 (2002).

Sommerfeld, A.

A. Sommerfeld, Optics, transl. by O. Laporte, and P. A. Moldauer (Academic, New York, 1954), Sec. 34.

Souli, N.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 9.

Sugita, A.

Y. Nishiyama, A. Sugita, M. Endo, H. Kogoe, K. Yorimasa, T. Sakai, T. Kurosawa, “Birefringence of a transparent cylinder determined by light scattering. Spider silk,” J. Fac. Edu. Hum. Sci. Yokohama Nat. Univ. 4, 11–17 (2002).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles, (Dover, New York, 1981), Chap. 15.

Watanabe, T.

Y. Nishiyama, S. Kurita, I. Yamamoto, Y. Ishizuka, T. Watanabe, D. Kobayashi, K. Odanaka, Y. Hyogo, “Diameter and refractive index of a cylindrical thread determined by scattered light pattern,” Opt. Rev. 8, 90–94 (2001).
[CrossRef]

Watkins, L. S.

Wolf, E.

M. Born, E. Wolf, Principles of Optics7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8.
[CrossRef]

Yamamoto, I.

Y. Nishiyama, S. Kurita, I. Yamamoto, Y. Ishizuka, T. Watanabe, D. Kobayashi, K. Odanaka, Y. Hyogo, “Diameter and refractive index of a cylindrical thread determined by scattered light pattern,” Opt. Rev. 8, 90–94 (2001).
[CrossRef]

Yorimasa, K.

Y. Nishiyama, A. Sugita, M. Endo, H. Kogoe, K. Yorimasa, T. Sakai, T. Kurosawa, “Birefringence of a transparent cylinder determined by light scattering. Spider silk,” J. Fac. Edu. Hum. Sci. Yokohama Nat. Univ. 4, 11–17 (2002).

Zimmermann, E.

Appl. Opt. (1)

Eur. J. Phys. (1)

L. Kirkup, “A guide to GUM,” Eur. J. Phys. 23, 483–487 (2002).
[CrossRef]

J. Colloid Interface Sci. (1)

J. L. Lundberg, “Light scattering from large fibers at normal incidence,” J. Colloid Interface Sci. 29, 565–583 (1969).
[CrossRef]

J. Fac. Edu. Hum. Sci. Yokohama Nat. Univ. (1)

Y. Nishiyama, A. Sugita, M. Endo, H. Kogoe, K. Yorimasa, T. Sakai, T. Kurosawa, “Birefringence of a transparent cylinder determined by light scattering. Spider silk,” J. Fac. Edu. Hum. Sci. Yokohama Nat. Univ. 4, 11–17 (2002).

J. Opt. Soc. Am. (3)

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

Opt. Rev. (1)

Y. Nishiyama, S. Kurita, I. Yamamoto, Y. Ishizuka, T. Watanabe, D. Kobayashi, K. Odanaka, Y. Hyogo, “Diameter and refractive index of a cylindrical thread determined by scattered light pattern,” Opt. Rev. 8, 90–94 (2001).
[CrossRef]

Other (9)

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 6.

M. Born, E. Wolf, Principles of Optics7th ed. (Cambridge U. Press, Cambridge, UK, 1999), Chap. 8.
[CrossRef]

A. Sommerfeld, Optics, transl. by O. Laporte, and P. A. Moldauer (Academic, New York, 1954), Sec. 34.

Ref. [2], Sec. 39.

Ref. [1], Chap. 11.

Ref. [2], Sec. 38.

H. C. van de Hulst, Light Scattering by Small Particles, (Dover, New York, 1981), Chap. 15.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), Chap. 9.

BIPM, IEC, IFCC, ISO, IUPAP and OIML, Guide to the Expression of Uncertainty in Measurement, 1st ed. (International Standards Organization, Geneva, 1993).

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Figures (6)

Fig. 1
Fig. 1

Photograph of a sample D 100 nm : (A) microscope, (B) scanning electron microscope.

Fig. 2
Fig. 2

Schematic of the system of measurement.

Fig. 3
Fig. 3

Illustration of the index of variation for the data A(a): (i) for a global region, (ii) for a local domain around the optimum point ( n 0 , D 0 ) .

Fig. 4
Fig. 4

Angular distribution of the scattering intensity for the sample (A) listed in Table 1. Circles or triangles are data measured for (a) perpendicular-polarized or (b) parallel-polarized wave. Solid curves show the optimum theoretical curves.

Fig. 5
Fig. 5

Same as Fig. 4 for the samples (B)–(E) listed in Table 1.

Fig. 6
Fig. 6

Approach of the angular distribution to dipole radiation (dashed line), illustrated for D λ 5 (solid curve) and λ 10 (double-dashed curve). ‖ or ⊥ shows the polarization of the scattered wave.

Tables (1)

Tables Icon

Table 1 Optimum Values of n or n and D, and the Uncertainties

Equations (16)

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ϵ 0 [ n 2 0 0 0 n 2 0 0 0 n 2 ] ,
σ α ( θ sc ) = 2 π k B α ( 0 ) + 2 m = 1 B α ( m ) cos ( m θ sc ) 2 ,
α = , ,
B ( m ) = J m ( n k r 0 ) J m + 1 ( k r 0 ) + n J m + 1 ( n k r 0 ) J m ( k r 0 ) J m ( n k r 0 ) H m + 1 ( 1 ) ( k r 0 ) n J m + 1 ( n k r 0 ) H m ( 1 ) ( k r 0 ) ,
B ( m ) = n J m ( n k r 0 ) J m ( k r 0 ) J m ( n k r 0 ) J m ( k r 0 ) n J m ( n k r 0 ) H m ( 1 ) ( k r 0 ) + J m ( n k r 0 ) H m ( 1 ) ( k r 0 ) .
J m ( z ) 1 m ! ( z 2 ) m ( 1 z 2 4 ( m + 1 ) ) , m 0 ,
Y m ( z ) ( m 1 ) ! π ( z 2 ) m , m 1 ,
Y 0 ( z ) 2 π ln ( z 2 ) .
B ( m ) i π ( n 2 1 ) m ! ( m + 1 ) ! ( k r 0 2 ) 2 m + 2 , m 0 .
σ ( θ sc ) 2 π k B ( 0 ) 2 = 2 π k ( n 2 1 ) 2 ( k r 0 2 ) 4 .
B ( 0 ) i π ( n 2 1 ) 2 ( k r 0 2 ) 4 ,
B ( m ) i π n m 1 ( n 2 1 ) m ! ( m 1 ) ! ( n 2 + 1 ) ( k r 0 2 ) 2 m , m 1 .
σ ( θ sc ) 8 π k ( n 2 1 ) 2 ( n 2 + 1 ) 2 ( k r 0 2 ) 4 [ cos θ sc + ( k r 0 2 ) 2 ( n 2 + 1 4 + n 2 cos 2 θ sc ) ] 2 .
U I ( n , D ) = { Σ i [ I i b σ i ( n , D ) ] 2 N } 1 2 ( Σ i I i N ) ,
search n min = min n G ̃ n , n max = max n G ̃ n ,
D min = min D G ̃ D , D max = max D G ̃ D .

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