Abstract

For the development of new kinds of refractive-index sensors for capillary electrophoresis, we investigated the scattering of an off-axis incident Gaussian beam by a homogeneous dielectric cylinder at normal incidence. The numerical calculations are based on the exact solution of the Helmholtz equation in circular cylindrical coordinates. Contrary to the procedures with geometrical and paraxial models, this procedure gives accurate results even when the beam dimensions are of the order of the wavelength of light and when the beam diameter is greater than the diameter of the dielectric cylinder. This rigorous electromagnetic treatment is verified by experimental measurements for cylinders with diameters from 5 to 100 μm.

© 1995 Optical Society of America

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References

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  1. A. G. Ewing, R. A. Wallingford, T. M. Oleforowicz, “Capillary electrophoresis,” Anal. Chem. 61, 292–303 (1991).
  2. B. Krattiger, A. E. Bruno, H. M. Widmer, M. Geiser, R. Dändliker, “Laser-based refractive-index detection for capillary electrophoresis: ray-tracing interference theory,” Appl. Opt. 32, 956–965 (1993).
    [Crossref] [PubMed]
  3. B. Krattiger, G. J. M. Bruin, A. E. Bruno, “Hologram-based refractive index detector for capillary electrophoresis: separation of metal ions,” Anal. Chem. 66, 1–8 (1994).
    [Crossref]
  4. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 15.
  5. W. A. Farone, C. W. Querfeld, “Electromagnetic scattering from radially inhomogeneous infinite cylinders at oblique incidence,”J. Opt. Soc. Am. 56, 476–480 (1966).
    [Crossref]
  6. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), Chap. 6.
  7. T. Kojima, Y. Yanagiuchi, “Scattering of an offset two-dimensional Gaussian beam wave by a cylinder,” J. Appl. Phys. 50, 41–46 (1979).
    [Crossref]
  8. S. Kosaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).
    [Crossref]
  9. S. Kozaki, “Scattering of a Gaussian beam by an inhomogeneous dielectric cylinder,”J. Opt. Soc. Am. 72, 1470–1474 (1982).
    [Crossref]
  10. A. Z. Elsherbeni, M. Hamid, G. Tian, “Iterative scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders,”J. Electro. Waves Appl. 7, 1323–1342 (1993).
    [Crossref]
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.
  12. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
    [Crossref]
  13. J. A. DeSanto, Scalar Wave Theory (Springer-Verlag, Berlin, 1992), Chap. 1.
    [Crossref]
  14. C. R. Wylie, L. C. Barrett, Advanced Engineering Mathematics, 5th ed. (McGraw-Hill, New York, 1985), Chap. 10.
  15. A. E. Bruno, B. Krattiger, F. Maystre, H. M Widmer, “On-column laser-based refractive-index detector for capillary electrophoresis,” Anal. Chem. 63, 2689–2697 (1991).
    [Crossref]

1994 (1)

B. Krattiger, G. J. M. Bruin, A. E. Bruno, “Hologram-based refractive index detector for capillary electrophoresis: separation of metal ions,” Anal. Chem. 66, 1–8 (1994).
[Crossref]

1993 (2)

B. Krattiger, A. E. Bruno, H. M. Widmer, M. Geiser, R. Dändliker, “Laser-based refractive-index detection for capillary electrophoresis: ray-tracing interference theory,” Appl. Opt. 32, 956–965 (1993).
[Crossref] [PubMed]

A. Z. Elsherbeni, M. Hamid, G. Tian, “Iterative scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders,”J. Electro. Waves Appl. 7, 1323–1342 (1993).
[Crossref]

1991 (2)

A. G. Ewing, R. A. Wallingford, T. M. Oleforowicz, “Capillary electrophoresis,” Anal. Chem. 61, 292–303 (1991).

A. E. Bruno, B. Krattiger, F. Maystre, H. M Widmer, “On-column laser-based refractive-index detector for capillary electrophoresis,” Anal. Chem. 63, 2689–2697 (1991).
[Crossref]

1982 (2)

S. Kosaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).
[Crossref]

S. Kozaki, “Scattering of a Gaussian beam by an inhomogeneous dielectric cylinder,”J. Opt. Soc. Am. 72, 1470–1474 (1982).
[Crossref]

1979 (2)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[Crossref]

T. Kojima, Y. Yanagiuchi, “Scattering of an offset two-dimensional Gaussian beam wave by a cylinder,” J. Appl. Phys. 50, 41–46 (1979).
[Crossref]

1966 (1)

Barrett, L. C.

C. R. Wylie, L. C. Barrett, Advanced Engineering Mathematics, 5th ed. (McGraw-Hill, New York, 1985), Chap. 10.

Bruin, G. J. M.

B. Krattiger, G. J. M. Bruin, A. E. Bruno, “Hologram-based refractive index detector for capillary electrophoresis: separation of metal ions,” Anal. Chem. 66, 1–8 (1994).
[Crossref]

Bruno, A. E.

B. Krattiger, G. J. M. Bruin, A. E. Bruno, “Hologram-based refractive index detector for capillary electrophoresis: separation of metal ions,” Anal. Chem. 66, 1–8 (1994).
[Crossref]

B. Krattiger, A. E. Bruno, H. M. Widmer, M. Geiser, R. Dändliker, “Laser-based refractive-index detection for capillary electrophoresis: ray-tracing interference theory,” Appl. Opt. 32, 956–965 (1993).
[Crossref] [PubMed]

A. E. Bruno, B. Krattiger, F. Maystre, H. M Widmer, “On-column laser-based refractive-index detector for capillary electrophoresis,” Anal. Chem. 63, 2689–2697 (1991).
[Crossref]

Dändliker, R.

DeSanto, J. A.

J. A. DeSanto, Scalar Wave Theory (Springer-Verlag, Berlin, 1992), Chap. 1.
[Crossref]

Elsherbeni, A. Z.

A. Z. Elsherbeni, M. Hamid, G. Tian, “Iterative scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders,”J. Electro. Waves Appl. 7, 1323–1342 (1993).
[Crossref]

Ewing, A. G.

A. G. Ewing, R. A. Wallingford, T. M. Oleforowicz, “Capillary electrophoresis,” Anal. Chem. 61, 292–303 (1991).

Farone, W. A.

Geiser, M.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.

Hamid, M.

A. Z. Elsherbeni, M. Hamid, G. Tian, “Iterative scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders,”J. Electro. Waves Appl. 7, 1323–1342 (1993).
[Crossref]

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[Crossref]

Kerker, M.

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

Kojima, T.

T. Kojima, Y. Yanagiuchi, “Scattering of an offset two-dimensional Gaussian beam wave by a cylinder,” J. Appl. Phys. 50, 41–46 (1979).
[Crossref]

Kosaki, S.

S. Kosaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).
[Crossref]

Kozaki, S.

Krattiger, B.

B. Krattiger, G. J. M. Bruin, A. E. Bruno, “Hologram-based refractive index detector for capillary electrophoresis: separation of metal ions,” Anal. Chem. 66, 1–8 (1994).
[Crossref]

B. Krattiger, A. E. Bruno, H. M. Widmer, M. Geiser, R. Dändliker, “Laser-based refractive-index detection for capillary electrophoresis: ray-tracing interference theory,” Appl. Opt. 32, 956–965 (1993).
[Crossref] [PubMed]

A. E. Bruno, B. Krattiger, F. Maystre, H. M Widmer, “On-column laser-based refractive-index detector for capillary electrophoresis,” Anal. Chem. 63, 2689–2697 (1991).
[Crossref]

Maystre, F.

A. E. Bruno, B. Krattiger, F. Maystre, H. M Widmer, “On-column laser-based refractive-index detector for capillary electrophoresis,” Anal. Chem. 63, 2689–2697 (1991).
[Crossref]

Oleforowicz, T. M.

A. G. Ewing, R. A. Wallingford, T. M. Oleforowicz, “Capillary electrophoresis,” Anal. Chem. 61, 292–303 (1991).

Querfeld, C. W.

Tian, G.

A. Z. Elsherbeni, M. Hamid, G. Tian, “Iterative scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders,”J. Electro. Waves Appl. 7, 1323–1342 (1993).
[Crossref]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 15.

Wallingford, R. A.

A. G. Ewing, R. A. Wallingford, T. M. Oleforowicz, “Capillary electrophoresis,” Anal. Chem. 61, 292–303 (1991).

Widmer, H. M

A. E. Bruno, B. Krattiger, F. Maystre, H. M Widmer, “On-column laser-based refractive-index detector for capillary electrophoresis,” Anal. Chem. 63, 2689–2697 (1991).
[Crossref]

Widmer, H. M.

Wylie, C. R.

C. R. Wylie, L. C. Barrett, Advanced Engineering Mathematics, 5th ed. (McGraw-Hill, New York, 1985), Chap. 10.

Yanagiuchi, Y.

T. Kojima, Y. Yanagiuchi, “Scattering of an offset two-dimensional Gaussian beam wave by a cylinder,” J. Appl. Phys. 50, 41–46 (1979).
[Crossref]

Am. J. Phys. (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[Crossref]

Anal. Chem. (3)

A. E. Bruno, B. Krattiger, F. Maystre, H. M Widmer, “On-column laser-based refractive-index detector for capillary electrophoresis,” Anal. Chem. 63, 2689–2697 (1991).
[Crossref]

A. G. Ewing, R. A. Wallingford, T. M. Oleforowicz, “Capillary electrophoresis,” Anal. Chem. 61, 292–303 (1991).

B. Krattiger, G. J. M. Bruin, A. E. Bruno, “Hologram-based refractive index detector for capillary electrophoresis: separation of metal ions,” Anal. Chem. 66, 1–8 (1994).
[Crossref]

Appl. Opt. (1)

J. Appl. Phys. (2)

T. Kojima, Y. Yanagiuchi, “Scattering of an offset two-dimensional Gaussian beam wave by a cylinder,” J. Appl. Phys. 50, 41–46 (1979).
[Crossref]

S. Kosaki, “Scattering of a Gaussian beam by a homogeneous dielectric cylinder,” J. Appl. Phys. 53, 7195–7200 (1982).
[Crossref]

J. Electro. Waves Appl. (1)

A. Z. Elsherbeni, M. Hamid, G. Tian, “Iterative scattering of a Gaussian beam by an array of circular conducting and dielectric cylinders,”J. Electro. Waves Appl. 7, 1323–1342 (1993).
[Crossref]

J. Opt. Soc. Am. (2)

Other (5)

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

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 15.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.

J. A. DeSanto, Scalar Wave Theory (Springer-Verlag, Berlin, 1992), Chap. 1.
[Crossref]

C. R. Wylie, L. C. Barrett, Advanced Engineering Mathematics, 5th ed. (McGraw-Hill, New York, 1985), Chap. 10.

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Figures (10)

Fig. 1
Fig. 1

Optical configurations for (a) the one-beam interferometer and (b) the two-beam interferometer.

Fig. 2
Fig. 2

Geometry for the scattering of an off-axis incident Gaussian beam by a homogeneous dielectric cylinder.

Fig. 3
Fig. 3

Geometry of the capillary illuminated by an off-axis incident Gaussian beam, which is focused in the plane z = −z0.

Fig. 4
Fig. 4

Experimental setup used to record the fringe patterns. RIMF, refractive-index matching fluid.

Fig. 5
Fig. 5

Fringe pattern for a 100-μm i.d. capillary illuminated by an off-axis beam with waist w0 = 8.4 μm [curve (a)]. The theoretical curves are calculated with a purely geometrical model2 [curve (b)] and a rigorous electromagnetic model [curve (c)]. Note that the intensity is represented on a logarithmic scale.

Fig. 6
Fig. 6

Measured and simulated fringe patterns for a 100-μm i.d. capillary with incident-beam waist w0 = 8.4 μm and a lateral beam shift y0 = 52.5 μm (intensity represented on a logarithmic scale). The measured pattern for positive scattering angles is the same as in Fig. 5.

Fig. 7
Fig. 7

Measured and simulated fringe patterns for a 25-μm i.d. capillary illuminated by an on-axis incident beam (y0 = 0) with waist w0 = 22 μm (intensity represented on a logarithmic scale).

Fig. 8
Fig. 8

Measured and simulated fringe patterns for a 25-μm i.d. capillary with incident-beam waist w0 = 22 μm and a lateral beam shift y0 = 21 μm (intensity represented on a logarithmic scale).

Fig. 9
Fig. 9

Measured and simulated fringe patterns for a 15-μm i.d. capillary with incident-beam waist w0 = 22 μm and a lateral beam shift y0 = 19 μm (intensity represented on a logarithmic scale).

Fig. 10
Fig. 10

Measured and simulated fringe patterns for a 5-μm i.d. capillary illuminated by an on-axis incident beam (y0 = 0) with a waist w0 = 22 μm (intensity represented on a logarithmic scale).

Equations (15)

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U x inc ( y , z ) = 1 2 π d α ^ A z 0 ( α ^ ) × exp [ - i n 1 2 k 2 - α ^ 2 ( z + z 0 ) - i α ^ y ] ,
A z 0 ( α ^ ) = d y U x inc ( y , - z 0 ) exp ( i α ^ y ) .
α ^ = 2 π λ 1 α = 2 π λ 1 cos ( π 2 - ϑ ) ,
U x inc ( y , - z 0 ) = U 0 exp [ - q ˜ 2 ( y - y 0 ) 2 ] ,
q ˜ = 1 w 2 + i n 1 k 2 R ,
U x inc ( r , ϑ ) = U 0 n = - ( - i ) n exp ( i n ϑ ) J n ( n 1 k r ) A n ,
A n = 1 2 π q ˜ d α ˜ exp { - α ^ 2 4 q ˜ 2 + i [ y 0 α ^ - n 1 2 k 2 - α ^ 2 z 0 - n γ ( α ^ ) ] } ,
sin γ = α ^ n 1 k .
U x sca ( r , ϑ ) = U 0 n = - ( - i ) n exp ( i n ϑ ) H n ( 2 ) ( n 1 k r ) B n ,
H m ( 1 ) , ( 2 ) ( n 1 k r ) 2 π λ r exp [ ± i ( n 1 k r - m π 2 - π 4 ) ]             when r .
U x out ( r , ϑ ) = U 0 n = - ( - i ) n exp ( i n ϑ ) [ J n ( n 1 k r ) A n + H n ( 2 ) ( n 1 k r ) B n ] ,
{ Y 0 ( n 2 k r ) log ( r ) Y m ( n 2 k r ) ( n 2 k r ) - m , m 0 }             when r 0 ,
U x int ( r , ϑ ) = U 0 n = - ( - i ) n exp ( i n ϑ ) J n ( n 2 k r ) C n .
B n = n 2 J n ( n 1 k a 0 ) J n ( n 2 k a 0 ) - n 1 J n ( n 2 k a 0 ) J n ( n 1 k a 0 ) n 2 H n ( 2 ) ( n 1 k a 0 ) J n ( n 2 k a 0 ) - n 1 J n ( n 2 k a 0 ) H n ( 2 ) ( n 1 k a 0 ) A n ,
C n = 2 i π k a 0 × A n n 2 H n ( 2 ) ( n 1 k a 0 ) J n ( n 2 k a 0 ) - n 1 J n ( n 2 k a 0 ) H n ( 2 ) ( n 1 k a 0 ) ,

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