Abstract

An extension of the problem of a moderately focused Gaussian beam scattered by an arrangement of two parallel nonabsorbing dielectric cylinders of arbitrary refraction indexes and radii is developed. The feature introduced in the present solution is the relative angular position coordinate between the two cylinders (ϕ0) as a degree of freedom. Explicit dependence on this variable of scattering coefficients, beam shape coefficient, and extinction efficiency is given. Together with the displacement coordinate d, various other scattering configurations can be studied. The applicability of the solution is given in the form of extinction curves taking ϕ0 as the main parameter for variation. The model is proposed as a means of investigating evanescent wave microscopy.

© 2008 Optical Society of America

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References

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  1. T. E. Starr and N. L. Thompson, “Total internal reflection with fluorescence correlation spectroscopy: combined surface reaction and solution diffusion,” Biophys. J. 80, 1575-1584(2001).
    [Crossref] [PubMed]
  2. D. Voigt, B. T. Wolschrijn, R. Jansen, B. Bhattacharya, R. J. C. Spreeuw, and H. V. L. Van den Heuvell, “Observation of radiation pressure exerted by evanescent waves,” Phys. Rev. A 61, 063412 (2000).
    [Crossref]
  3. P. C. Odiachi and D. C. Prieve, “Total internal reflection microscopy: distortion caused by additive noise,” Ind. Eng. Chem. Res. 41, 478-485 (2002).
    [Crossref]
  4. M. A. S. Vigeant, M. Wagner, L. K. Tamm, and R. M. Ford, “Nanometer distances between swimming bacteria and surfaces measured by total internal reflection aqueous fluorescence microscopy,” Langmuir 17, 2235-2242 (2001).
    [Crossref]
  5. V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticles on a surface,” Appl. Phys. Lett. 86, 0311061 (2005).
    [Crossref]
  6. T. Tanaka and S. Yamamoto, “Optically induced propulsion of small particles in a evanescent field of higher propagation mode in a multimode channeled waveguide,” Appl. Phys. Lett. 77, 3131-3133 (2000).
    [Crossref]
  7. A. G. Simão, L. G. Guimarães, and J. P. R. F. de Mendonça, “Evanescent wave coupling in light scattering of an off-axis normally incident Gaussian beam by two parallel non-absorbing cylinders,” J. Opt. Soc. Am. A 19, 2053-2063 (2002).
    [Crossref]
  8. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  9. J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A. 14, 640-652 (1997).
    [Crossref]
  10. D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526-2538(1994).
    [Crossref]
  11. J. A. Lock, J. T. Hodges, and G. Goesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. A. 12, 2708-2714 (1995).
    [Crossref]
  12. S. Lee, “Scattering of polarized radiation by an arbitrary collection of closely spaced parallel nonhomogeneous tilted cylinders,” J. Opt. Soc. Am. A 13, 2256-2265 (1996).
    [Crossref]
  13. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  14. A. Cohen and P. Alpert, “Extinction efficiency of obliquely and randomly oriented infinite cylinders,” J. Appl. Phys. 50, 8262-8264 (1979).
    [Crossref]
  15. L. G. Guimarães, “Theory of Mie caustics,” Opt. Commun. 103, 339-344 (1993).
    [Crossref]
  16. A. E. Walsby, “Gas vesicles,” Microbiol. Rev. 58, 94-144 (1994).
    [PubMed]

2005 (1)

V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticles on a surface,” Appl. Phys. Lett. 86, 0311061 (2005).
[Crossref]

2002 (2)

2001 (2)

M. A. S. Vigeant, M. Wagner, L. K. Tamm, and R. M. Ford, “Nanometer distances between swimming bacteria and surfaces measured by total internal reflection aqueous fluorescence microscopy,” Langmuir 17, 2235-2242 (2001).
[Crossref]

T. E. Starr and N. L. Thompson, “Total internal reflection with fluorescence correlation spectroscopy: combined surface reaction and solution diffusion,” Biophys. J. 80, 1575-1584(2001).
[Crossref] [PubMed]

2000 (2)

D. Voigt, B. T. Wolschrijn, R. Jansen, B. Bhattacharya, R. J. C. Spreeuw, and H. V. L. Van den Heuvell, “Observation of radiation pressure exerted by evanescent waves,” Phys. Rev. A 61, 063412 (2000).
[Crossref]

T. Tanaka and S. Yamamoto, “Optically induced propulsion of small particles in a evanescent field of higher propagation mode in a multimode channeled waveguide,” Appl. Phys. Lett. 77, 3131-3133 (2000).
[Crossref]

1997 (1)

J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A. 14, 640-652 (1997).
[Crossref]

1996 (1)

1995 (1)

J. A. Lock, J. T. Hodges, and G. Goesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. A. 12, 2708-2714 (1995).
[Crossref]

1994 (2)

1993 (1)

L. G. Guimarães, “Theory of Mie caustics,” Opt. Commun. 103, 339-344 (1993).
[Crossref]

1979 (1)

A. Cohen and P. Alpert, “Extinction efficiency of obliquely and randomly oriented infinite cylinders,” J. Appl. Phys. 50, 8262-8264 (1979).
[Crossref]

Alpert, P.

A. Cohen and P. Alpert, “Extinction efficiency of obliquely and randomly oriented infinite cylinders,” J. Appl. Phys. 50, 8262-8264 (1979).
[Crossref]

Bhattacharya, B.

D. Voigt, B. T. Wolschrijn, R. Jansen, B. Bhattacharya, R. J. C. Spreeuw, and H. V. L. Van den Heuvell, “Observation of radiation pressure exerted by evanescent waves,” Phys. Rev. A 61, 063412 (2000).
[Crossref]

Cohen, A.

A. Cohen and P. Alpert, “Extinction efficiency of obliquely and randomly oriented infinite cylinders,” J. Appl. Phys. 50, 8262-8264 (1979).
[Crossref]

de Mendonça, J. P. R. F.

Dholakia, K.

V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticles on a surface,” Appl. Phys. Lett. 86, 0311061 (2005).
[Crossref]

Felbacq, D.

Ford, R. M.

M. A. S. Vigeant, M. Wagner, L. K. Tamm, and R. M. Ford, “Nanometer distances between swimming bacteria and surfaces measured by total internal reflection aqueous fluorescence microscopy,” Langmuir 17, 2235-2242 (2001).
[Crossref]

Garcés-Chávez, V.

V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticles on a surface,” Appl. Phys. Lett. 86, 0311061 (2005).
[Crossref]

Goesbet, G.

J. A. Lock, J. T. Hodges, and G. Goesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. A. 12, 2708-2714 (1995).
[Crossref]

Guimarães, L. G.

Hodges, J. T.

J. A. Lock, J. T. Hodges, and G. Goesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. A. 12, 2708-2714 (1995).
[Crossref]

Jansen, R.

D. Voigt, B. T. Wolschrijn, R. Jansen, B. Bhattacharya, R. J. C. Spreeuw, and H. V. L. Van den Heuvell, “Observation of radiation pressure exerted by evanescent waves,” Phys. Rev. A 61, 063412 (2000).
[Crossref]

Lee, S.

Lock, J. A.

J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A. 14, 640-652 (1997).
[Crossref]

J. A. Lock, J. T. Hodges, and G. Goesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. A. 12, 2708-2714 (1995).
[Crossref]

Maystre, D.

Odiachi, P. C.

P. C. Odiachi and D. C. Prieve, “Total internal reflection microscopy: distortion caused by additive noise,” Ind. Eng. Chem. Res. 41, 478-485 (2002).
[Crossref]

Prieve, D. C.

P. C. Odiachi and D. C. Prieve, “Total internal reflection microscopy: distortion caused by additive noise,” Ind. Eng. Chem. Res. 41, 478-485 (2002).
[Crossref]

Simão, A. G.

Spalding, G. C.

V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticles on a surface,” Appl. Phys. Lett. 86, 0311061 (2005).
[Crossref]

Spreeuw, R. J. C.

D. Voigt, B. T. Wolschrijn, R. Jansen, B. Bhattacharya, R. J. C. Spreeuw, and H. V. L. Van den Heuvell, “Observation of radiation pressure exerted by evanescent waves,” Phys. Rev. A 61, 063412 (2000).
[Crossref]

Starr, T. E.

T. E. Starr and N. L. Thompson, “Total internal reflection with fluorescence correlation spectroscopy: combined surface reaction and solution diffusion,” Biophys. J. 80, 1575-1584(2001).
[Crossref] [PubMed]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Tamm, L. K.

M. A. S. Vigeant, M. Wagner, L. K. Tamm, and R. M. Ford, “Nanometer distances between swimming bacteria and surfaces measured by total internal reflection aqueous fluorescence microscopy,” Langmuir 17, 2235-2242 (2001).
[Crossref]

Tanaka, T.

T. Tanaka and S. Yamamoto, “Optically induced propulsion of small particles in a evanescent field of higher propagation mode in a multimode channeled waveguide,” Appl. Phys. Lett. 77, 3131-3133 (2000).
[Crossref]

Tayeb, G.

Thompson, N. L.

T. E. Starr and N. L. Thompson, “Total internal reflection with fluorescence correlation spectroscopy: combined surface reaction and solution diffusion,” Biophys. J. 80, 1575-1584(2001).
[Crossref] [PubMed]

van de Hulst, H. C.

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

Van den Heuvell, H. V. L.

D. Voigt, B. T. Wolschrijn, R. Jansen, B. Bhattacharya, R. J. C. Spreeuw, and H. V. L. Van den Heuvell, “Observation of radiation pressure exerted by evanescent waves,” Phys. Rev. A 61, 063412 (2000).
[Crossref]

Vigeant, M. A. S.

M. A. S. Vigeant, M. Wagner, L. K. Tamm, and R. M. Ford, “Nanometer distances between swimming bacteria and surfaces measured by total internal reflection aqueous fluorescence microscopy,” Langmuir 17, 2235-2242 (2001).
[Crossref]

Voigt, D.

D. Voigt, B. T. Wolschrijn, R. Jansen, B. Bhattacharya, R. J. C. Spreeuw, and H. V. L. Van den Heuvell, “Observation of radiation pressure exerted by evanescent waves,” Phys. Rev. A 61, 063412 (2000).
[Crossref]

Wagner, M.

M. A. S. Vigeant, M. Wagner, L. K. Tamm, and R. M. Ford, “Nanometer distances between swimming bacteria and surfaces measured by total internal reflection aqueous fluorescence microscopy,” Langmuir 17, 2235-2242 (2001).
[Crossref]

Walsby, A. E.

A. E. Walsby, “Gas vesicles,” Microbiol. Rev. 58, 94-144 (1994).
[PubMed]

Wolschrijn, B. T.

D. Voigt, B. T. Wolschrijn, R. Jansen, B. Bhattacharya, R. J. C. Spreeuw, and H. V. L. Van den Heuvell, “Observation of radiation pressure exerted by evanescent waves,” Phys. Rev. A 61, 063412 (2000).
[Crossref]

Yamamoto, S.

T. Tanaka and S. Yamamoto, “Optically induced propulsion of small particles in a evanescent field of higher propagation mode in a multimode channeled waveguide,” Appl. Phys. Lett. 77, 3131-3133 (2000).
[Crossref]

Appl. Phys. Lett. (2)

V. Garcés-Chávez, K. Dholakia, and G. C. Spalding, “Extended-area optically induced organization of microparticles on a surface,” Appl. Phys. Lett. 86, 0311061 (2005).
[Crossref]

T. Tanaka and S. Yamamoto, “Optically induced propulsion of small particles in a evanescent field of higher propagation mode in a multimode channeled waveguide,” Appl. Phys. Lett. 77, 3131-3133 (2000).
[Crossref]

Biophys. J. (1)

T. E. Starr and N. L. Thompson, “Total internal reflection with fluorescence correlation spectroscopy: combined surface reaction and solution diffusion,” Biophys. J. 80, 1575-1584(2001).
[Crossref] [PubMed]

Ind. Eng. Chem. Res. (1)

P. C. Odiachi and D. C. Prieve, “Total internal reflection microscopy: distortion caused by additive noise,” Ind. Eng. Chem. Res. 41, 478-485 (2002).
[Crossref]

J. Appl. Phys. (1)

A. Cohen and P. Alpert, “Extinction efficiency of obliquely and randomly oriented infinite cylinders,” J. Appl. Phys. 50, 8262-8264 (1979).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. A. (2)

J. A. Lock, J. T. Hodges, and G. Goesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. A. 12, 2708-2714 (1995).
[Crossref]

J. A. Lock, “Scattering of a diagonally incident focused Gaussian beam by an infinitely long homogeneous circular cylinder,” J. Opt. Soc. Am. A. 14, 640-652 (1997).
[Crossref]

Langmuir (1)

M. A. S. Vigeant, M. Wagner, L. K. Tamm, and R. M. Ford, “Nanometer distances between swimming bacteria and surfaces measured by total internal reflection aqueous fluorescence microscopy,” Langmuir 17, 2235-2242 (2001).
[Crossref]

Microbiol. Rev. (1)

A. E. Walsby, “Gas vesicles,” Microbiol. Rev. 58, 94-144 (1994).
[PubMed]

Opt. Commun. (1)

L. G. Guimarães, “Theory of Mie caustics,” Opt. Commun. 103, 339-344 (1993).
[Crossref]

Phys. Rev. A (1)

D. Voigt, B. T. Wolschrijn, R. Jansen, B. Bhattacharya, R. J. C. Spreeuw, and H. V. L. Van den Heuvell, “Observation of radiation pressure exerted by evanescent waves,” Phys. Rev. A 61, 063412 (2000).
[Crossref]

Other (2)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

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

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Figures (7)

Fig. 1
Fig. 1

Scattering geometry of the two nonabsorbing dielectric cylinders. The probe cylinder can occupy position in the space given by coordinates ( d , ϕ 0 ) .

Fig. 2
Fig. 2

Behavior of the extinction efficiency for the Gaussian beam scattering of a metallic cylinder for different values of the beam waist parameter in units of the wavelength. Departure from the plane wave result in the strip 10 > β > 1 for values of Ω < 5 is clearly shown.

Fig. 3
Fig. 3

Behavior of the extinction efficiency for ϕ 0 variation for β = 10 on-axis incidence on the pump cylinder, which is in the center, N 1 / N 0 = 1 , N 2 / N 0 = 1.55 , and Ω = 5 . As a test of symmetry, the probe cylinder occupies the following coordinates: ( d = 3 ; ϕ 0 = 0 ° ) , ( d = 3 ; ϕ 0 = 90 ° ) , ( d = 3 , ϕ 0 = 180 ° ) , and ( d = 3 ; ϕ 0 = 270 ° ) . The inset shows the values of Q ext on two head-on positions, where at 90 ° , the Gaussian beam is convergent, and at 270 ° , the Gaussian beam is divergent.

Fig. 4
Fig. 4

Extinction factor for the Gaussian beam scattering by two nonabsorbing cylinders for two different angular positions for various values of separation distances d / a , while their refraction index, beam waist position coordinates, and size are kept fixed at the values 1.46, ( x 0 = y 0 = z 0 = 0 ) , and Ω = 5 , respectively. The incidence of the Gaussian beam occurs at the pump cylinder, and the probe is positioned at ϕ = 0 ° and 90 ° . The oscillations are out of phase between the two geometries.

Fig. 5
Fig. 5

Extinction factor for the Gaussian beam scattering by two nonabsorbing cylinders for the same angular position ϕ = 0 ° and β = 10 . The refraction indexes of both cylinders are varied for N 1 = N 2 = 1.55 and N 1 = N 2 = 1.46 . Again the incidence of the Gaussian beam is on axis on the pump cylinder. The size of the beam waist is Ω = 5 .

Fig. 6
Fig. 6

Extinction factor for angular position variation of 0 ϕ 0 180 ° . The parameters used in this plot were β = 17.4983 , Ω = 5 , n = 23 , and a 1 / a 2 = 1.5 . The square dots correspond to data points obtained for the rotation of the probe out of the external caustic region of the pump, thus d / a 1 = 2.62 . The inset shows the actual position and size of the cylinders for ϕ 0 = 0 ° , 90 ° , and 180 ° in the ( y x ) plane. The dashed arrow indicates the beam’s direction, and the focal waist center position is the small cross in the same line. The dotted circle is the beam waist at focal plane in ( z y ) . It illustrates the extension of the focal spot over the cylinders.

Fig. 7
Fig. 7

Sensitivity of the extinction factor on resonance regime for angular position variation of 0 ϕ 0 180 ° . The parameters used in this plot were β = 17.4983 , Ω = 5 , n = 23 , and a 1 / a 2 = 2 . The dots correspond to data points obtained for the rotation of the probe in the external caustic region of the pump. At values of ϕ 0 < 90 ° , the probe scatters mainly evanescent waves generated by the pump. The distance between the cylinders for this situation is d / a 1 = 2 . The inset shows the actual position and size of the cylinders for ϕ 0 = 0 ° , 90 ° , and 180 ° in the ( y x ) plane. The dashed arrow indicates the beam’s direction, and the small cross in the same line is the focal spot center position. The beam waist at focal spot in the ( z y ) plane is represented by the dotted circle.

Equations (22)

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w 0 = 2 π Ω a 1 β 1 ,
Ψ Total μ = Ψ inc μ + Ψ scatt _ 1 μ + Ψ scatt _ 2 μ ,
Ψ inc μ = + N 0 d h n = F n A n ( O 1 , O 2 ) μ J n ( ρ 1 , 2 k 0 N 0 2 h 2 ) e i n ϕ 1 , 2 e i h k 0 z 1 , 2 ,
Ψ int μ , O j = + N j d h n = F n a n , O j μ J n ( ρ j k 0 N j 2 h 2 ) e i n ϕ j e i h k 0 z j ,
Ψ scatt μ , O j = + N 0 d h n = F n A n , O j s c , μ H n ( 1 ) ( ρ j k 0 N 0 2 h 2 ) e i n ϕ j e i h k 0 z j ,
H l ( 1 ) ( k 0 ρ 2 ) e i l ϕ 2 = q = q = H q l ( 1 ) ( k 0 d ) J q ( k 0 ρ 1 ) e i q ϕ 1 e i ( l q ) ϕ 0 .
H l ( 1 ) ( k 0 ρ 2 ) e i l ϕ 2 = q = q = J q l ( 1 ) ( k 0 d ) H q ( 1 ) ( k 0 ρ 1 ) e i q ϕ 1 e i ( l q ) ϕ 0 .
H l ( 1 ) ( k ˜ ρ 2 ) e i l ϕ 2 = q H q - l ( 1 ) ( k ˜ d ) J q ( k ˜ ρ 1 ) e i q ϕ 1 e i ( l - q ) ϕ 0 ,
k = { δ n , k T n O 2 q = F q F k H q k ( 1 ) ( w ) H q n ( 1 ) ( w ) T q O 1 e i ( k + n 2 q ) ϕ 0 } A k , O 2 s c , μ = F n A n , O 2 μ T n O 2 + T n O 2 q = ( F q ) 2 A q , O 1 μ T q O 1 H q n ( 1 ) ( w ) e i ( n q ) ϕ 0 e i k 0 h ( z 1 z 2 ) ,
T n O j = ( 2 i ( H n ( u j ) u j H n ( u j ) μ j μ 0 J n ( v j ) v j J n ( v j ) ) u j 2 π [ H n ( u j ) ] 2 { D 1 j D 2 j ( n h ) 2 ( 1 u j 2 1 v j 2 ) 2 } J n ( u j ) H n ( u j ) ) ,
u j = a j k 0 N 0 2 h 2 ,
v j = a j k 0 N j 2 h 2 ,
w = d k 0 N 0 2 h 2 ,
D 1 j = ( H n ( u j ) u j H n ( u j ) μ j μ 0 J n ( v j ) v j J n ( v j ) ) ,
D 2 j = ( H n ( u j ) u j H n ( u j ) ϵ j ϵ 0 J n ( v j ) v j J n ( v j ) ) .
C s c 4 k 0 n = s = e i ( n s ) π / 2 J s n ( k 0 d ) [ A n , O 1 s c , μ A n , O 2 s c , μ * + A n , O 2 s c , μ A n , O 1 s c , μ * ] + n = s = e i ( n s ) ( ϕ 0 π / 2 ) | A n , O 1 s c , μ | 2 + n = s = e i ( n s ) ( ϕ 0 π / 2 ) | A n , O 2 s c , μ | 2 .
A n , O 2 μ = 1 2 s π 1 / 2 × exp [ i k 0 d sin ϕ 0 ] [ 1 2 i s 2 k 0 d sin ϕ 0 ] 1 / 2 × exp { s 2 [ n + k 0 d cos ϕ 0 ] 2 1 2 i s 2 k 0 d sin ϕ 0 } ,
C g = 2 β k 0 [ 1 + cos ϕ 0 a 2 a 1 ( 1 e Ω ) ] ,
Q ext = C s c C g .
β N d a = m π ,
Δ N N = Δ D D .
Δ N N = 0.06.

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