Abstract

The scattering of plane waves by dielectric particles is an age-old problem for which a complete solution was given by Gustav Mie (1908). Mie’s solution to the plane-wave case was later extended to the evanescent case in order to achieve resolutions beyond the Rayleigh limit. Solutions exist based on the multipole expansion method and group-theory method. Present work suggests an alternative solution to the scattering of evanescent waves by a spherical dielectric particle, by obtaining the scattering coefficients from Debye’s potentials as solved by Born and Wolf in the plane-wave case.

© 2009 Optical Society of America

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References

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  1. G. Mie, “Contibutions to the optics of turbid media, particularly of colloidal metal solutions,” Ann. Phys. 25, 377-445 (1908).
    [CrossRef]
  2. H. C. van de Hulst, Light Scattering by Small Particles (Dover Publications, 1981).
  3. H. Chew, D. S. Wang, and M. Kerker, “Elastic scattering of evanescent electromagnetic waves,” Appl. Opt. 18, 2679-2687 (1979).
    [CrossRef] [PubMed]
  4. C. Liu, T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521-531 (1995).
    [CrossRef]
  5. A. V. Zvyagin and K. Goto, “Mie scattering of evanescent waves by a dielectric sphere: comparison of multipole expansion method and group-theory methods,” J. Opt. Soc. Am. A 15, 3003-3008 (1998).
    [CrossRef]
  6. K. Belkebir, P. C. Chaumet, and A. Sentenac, “Super resolution in total internal reflection tomography,” J. Opt. Soc. Am. A 22, 1889-1897 (2005).
    [CrossRef]
  7. M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).
  8. C. A. Sciammarella, “Experimental mechanics at the nanometric level,” STRAIN 44, 3-19 (2008).
    [CrossRef]
  9. D. Axelrod, E. H. Hellen, and R. M. Fulbright, “Total internal reflection flourescence,” in Topics in Fluorescence Spectroscopy: Biochemical Applications, J.R.Lakowicz, ed. (Plenum Press, 1992), pp. 289-343.
  10. G. T. di Francia, La Diffrazione della Luce (Edizioni Scientifiche Einaudi, 1958).
  11. J. M. Guerra, “Super-resolution through illumination by diffraction-born evanescent waves,” Appl. Phys. Lett. 66, 3555-3557 (1995).
    [CrossRef]
  12. A. Ranfagni, D. Mugnai, and R. Ruggeri, “Beyond the diffraction limit: Super-resolving pupils,” J. Appl. Phys. 95, 2217-2222 (2004).
    [CrossRef]

2008 (1)

C. A. Sciammarella, “Experimental mechanics at the nanometric level,” STRAIN 44, 3-19 (2008).
[CrossRef]

2005 (1)

2004 (1)

A. Ranfagni, D. Mugnai, and R. Ruggeri, “Beyond the diffraction limit: Super-resolving pupils,” J. Appl. Phys. 95, 2217-2222 (2004).
[CrossRef]

1998 (1)

1995 (2)

C. Liu, T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521-531 (1995).
[CrossRef]

J. M. Guerra, “Super-resolution through illumination by diffraction-born evanescent waves,” Appl. Phys. Lett. 66, 3555-3557 (1995).
[CrossRef]

1979 (1)

1908 (1)

G. Mie, “Contibutions to the optics of turbid media, particularly of colloidal metal solutions,” Ann. Phys. 25, 377-445 (1908).
[CrossRef]

Axelrod, D.

D. Axelrod, E. H. Hellen, and R. M. Fulbright, “Total internal reflection flourescence,” in Topics in Fluorescence Spectroscopy: Biochemical Applications, J.R.Lakowicz, ed. (Plenum Press, 1992), pp. 289-343.

Belkebir, K.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).

Chaumet, P. C.

Chew, H.

di Francia, G. T.

G. T. di Francia, La Diffrazione della Luce (Edizioni Scientifiche Einaudi, 1958).

Fulbright, R. M.

D. Axelrod, E. H. Hellen, and R. M. Fulbright, “Total internal reflection flourescence,” in Topics in Fluorescence Spectroscopy: Biochemical Applications, J.R.Lakowicz, ed. (Plenum Press, 1992), pp. 289-343.

Goto, K.

Guerra, J. M.

J. M. Guerra, “Super-resolution through illumination by diffraction-born evanescent waves,” Appl. Phys. Lett. 66, 3555-3557 (1995).
[CrossRef]

Hellen, E. H.

D. Axelrod, E. H. Hellen, and R. M. Fulbright, “Total internal reflection flourescence,” in Topics in Fluorescence Spectroscopy: Biochemical Applications, J.R.Lakowicz, ed. (Plenum Press, 1992), pp. 289-343.

Kaiser, T.

C. Liu, T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521-531 (1995).
[CrossRef]

Kerker, M.

Lange, S.

C. Liu, T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521-531 (1995).
[CrossRef]

Liu, C.

C. Liu, T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521-531 (1995).
[CrossRef]

Mie, G.

G. Mie, “Contibutions to the optics of turbid media, particularly of colloidal metal solutions,” Ann. Phys. 25, 377-445 (1908).
[CrossRef]

Mugnai, D.

A. Ranfagni, D. Mugnai, and R. Ruggeri, “Beyond the diffraction limit: Super-resolving pupils,” J. Appl. Phys. 95, 2217-2222 (2004).
[CrossRef]

Ranfagni, A.

A. Ranfagni, D. Mugnai, and R. Ruggeri, “Beyond the diffraction limit: Super-resolving pupils,” J. Appl. Phys. 95, 2217-2222 (2004).
[CrossRef]

Ruggeri, R.

A. Ranfagni, D. Mugnai, and R. Ruggeri, “Beyond the diffraction limit: Super-resolving pupils,” J. Appl. Phys. 95, 2217-2222 (2004).
[CrossRef]

Schweiger, G.

C. Liu, T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521-531 (1995).
[CrossRef]

Sciammarella, C. A.

C. A. Sciammarella, “Experimental mechanics at the nanometric level,” STRAIN 44, 3-19 (2008).
[CrossRef]

Sentenac, A.

van de Hulst, H. C.

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

Wang, D. S.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).

Zvyagin, A. V.

Ann. Phys. (1)

G. Mie, “Contibutions to the optics of turbid media, particularly of colloidal metal solutions,” Ann. Phys. 25, 377-445 (1908).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. M. Guerra, “Super-resolution through illumination by diffraction-born evanescent waves,” Appl. Phys. Lett. 66, 3555-3557 (1995).
[CrossRef]

J. Appl. Phys. (1)

A. Ranfagni, D. Mugnai, and R. Ruggeri, “Beyond the diffraction limit: Super-resolving pupils,” J. Appl. Phys. 95, 2217-2222 (2004).
[CrossRef]

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

Opt. Commun. (1)

C. Liu, T. Kaiser, S. Lange, and G. Schweiger, “Structural resonances in a dielectric sphere illuminated by an evanescent wave,” Opt. Commun. 117, 521-531 (1995).
[CrossRef]

STRAIN (1)

C. A. Sciammarella, “Experimental mechanics at the nanometric level,” STRAIN 44, 3-19 (2008).
[CrossRef]

Other (4)

D. Axelrod, E. H. Hellen, and R. M. Fulbright, “Total internal reflection flourescence,” in Topics in Fluorescence Spectroscopy: Biochemical Applications, J.R.Lakowicz, ed. (Plenum Press, 1992), pp. 289-343.

G. T. di Francia, La Diffrazione della Luce (Edizioni Scientifiche Einaudi, 1958).

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).

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

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Figures (5)

Fig. 1
Fig. 1

Normalized intensity distribution for spheres of various diameters.

Fig. 2
Fig. 2

Experimental setup.

Fig. 3
Fig. 3

Experimental results of scattering of evanescent waves [8].

Fig. 4
Fig. 4

Normalized intensity distribution by a sphere of diameter 6 μ m .

Fig. 5
Fig. 5

The diffraction pattern of a three-coronae pupil, as given by the above equation for n = 3 (solid curve), and that of normal pupil (dashed curve) [12].

Equations (35)

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E p = E p ( 0 ) E p i ( α x i ̂ + α z k ̂ ) e i x k ( I ) sin Ω e z d ,
H p = E p ( 0 ) E p i ( α y j ̂ ) e i x k ( I ) sin Ω e z d ,
d = λ 0 2 π n 1 2 sin 2 Ω n 2 2 ,
E p ( 0 ) = 2 cos Ω [ 2 sin 2 Ω n 2 ] 1 2 n 2 cos Ω + i [ sin 2 Ω n 2 ] 1 2 ,
α x = i ( sin 2 Ω n 2 2 sin 2 Ω n 2 ) 1 2 ,
α y = n n 2 ( 2 sin 2 Ω n 2 ) 1 2 ,
α z = sin Ω ( 2 sin 2 Ω n 2 ) 1 2 .
{ E r = E r e + E r m = 2 ( r Π e ) r 2 + k 2 r Π e E θ = E θ e + E θ m = 1 r 2 ( r Π e ) r θ + k 2 r sin θ ( r Π e ) ϕ E ϕ = E ϕ e + E ϕ m = 1 r sin θ 2 ( r Π e ) r ϕ k 2 r ( r Π m ) θ H r = H r m + H r e = k 2 r Π m + 2 ( r Π m ) r 2 H θ = H θ m + H θ e = k 1 r sin θ ( r Π e ) ϕ + 1 r 2 ( r Π m ) r θ H ϕ = H ϕ m + H ϕ e = k 1 r ( r Π e ) θ + 1 r sin θ 2 ( r Π m ) r ϕ }
k 1 = i ω c [ ϵ + i 4 π σ ω ] , k 2 = i ω c ,
r Π = r l = 0 m = l l Π l ( m ) = l = 0 m = l l [ c l ψ l ( k r ) + d l χ l ( k r ) ] [ P l ( m ) ( cos θ ) ] [ a m cos ( m ϕ ) + b m sin ( m ϕ ) ] ,
A p x e sin θ e r cos θ d e i k ( I ) r sin θ cos ϕ sin Ω = l = 0 α l x k ( I ) 2 r 2 l ( l + 1 ) ψ l ( k ( I ) r ) P l ( 1 ) ( cos θ ) .
α l x = A p x e k ( I ) 2 a 2 A q x 0 π sin 2 θ e a cos θ d J 0 ( k ( I ) a sin θ sin Ω ) P q ( 1 ) ( cos θ ) d θ .
α l y = A p y m k ( I ) 2 a 2 A q y 0 π sin 2 θ e a cos θ d J 0 ( k ( I ) a sin θ sin Ω ) P q ( 1 ) ( cos θ ) d θ ,
α l z = A p z e k ( I ) 2 a 2 A q z 0 π sin θ cos θ e a cos θ d J 0 ( k ( I ) a sin θ sin Ω ) P q ( cos θ ) d θ .
r [ r Π ( i ) e + r Π ( s ) e ] r = a = r [ r Π ( w ) e ] r = a ,
r [ r Π ( i ) m + r Π ( s ) m ] r = a = r [ r Π ( w ) m ] r = a ,
k 1 ( I ) [ r Π ( i ) e + r Π ( s ) e ] r = a = k 1 ( II ) [ r Π ( w ) e ] r = a ,
k 2 ( I ) [ r Π ( i ) m + r Π ( s ) m ] r = a = k 2 ( II ) [ r Π ( w ) m ] r = a .
r Π ( w ) e = 1 k ( II ) 2 l = 0 ψ l ( k ( II ) r ) [ A l x P l ( 1 ) ( cos θ ) cos ϕ + A l z P l ( cos θ ) ] ,
r Π ( w ) m = 1 k ( II ) 2 l = 0 A l y ψ l ( k ( II ) r ) P l ( 1 ) ( cos θ ) sin ϕ ,
r Π ( s ) e = 1 k ( I ) 2 l = 0 ζ l ( 1 ) ( k ( I ) r ) [ B l x P l ( 1 ) ( cos θ ) cos ϕ + B l z P l ( cos θ ) ] ,
r Π ( s ) m = 1 k ( I ) 2 l = 0 B l y ζ l ( 1 ) ( k ( I ) r ) P l ( 1 ) ( cos θ ) sin ϕ .
B l x = α l x [ k 1 ( I ) ψ l ( k ( II ) a ) ψ l ( k ( I ) a ) k 1 ( II ) ψ l ( k ( II ) a ) ψ l ( k ( I ) a ) ] k 1 ( II ) ψ l ( k ( II ) a ) ζ l ( 1 ) ( k ( I ) a ) k 1 ( I ) ψ l ( k ( II ) a ) ζ l ( 1 ) ( k ( I ) a ) ,
B l y = α l y [ k 2 ( I ) ψ l ( k ( II ) a ) ψ l ( k ( I ) a ) k 2 ( II ) ψ l ( k ( II ) a ) ψ l ( k ( I ) a ) ] k 2 ( II ) ψ l ( k ( II ) a ) ζ l ( 1 ) ( k ( I ) a ) k 2 ( I ) ψ l ( k ( II ) a ) ζ l ( 1 ) ( k ( I ) a ) ,
B l z = α l z [ k 1 ( I ) ψ l ( k ( II ) a ) ψ l ( k ( I ) a ) k 1 ( II ) ψ l ( k ( II ) a ) ψ l ( k ( I ) a ) ] k 1 ( II ) ψ l ( k ( II ) a ) ζ l ( 1 ) ( k ( I ) a ) k 1 ( I ) ψ l ( k ( II ) a ) ζ l ( 1 ) ( k ( I ) a ) .
E r ( s ) = 2 ( r Π ( s ) e ) r 2 + k 2 r Π ( s ) e = 1 k ( I ) 2 r 2 l = 0 [ B l x P l ( 1 ) ( cos θ ) cos ϕ + B l z P l ( cos θ ) ] l ( l + 1 ) ζ l ( 1 ) ( k ( I ) r ) .
E θ ( s ) = 1 r 2 ( r Π ( s ) e ) r θ + k 2 ( I ) r sin θ ( r Π ( s ) m ) ϕ = 1 k ( I ) l = 0 [ k 2 ( I ) k ( I ) B l y ζ l ( 1 ) ( k ( I ) r ) P l ( 1 ) ( cos θ ) cos ϕ r sin θ [ B l x P l ( 1 ) ( cos θ ) cos ϕ + B l z P l ( cos θ ) ] ζ l ( 1 ) ( k ( I ) r ) sin θ r ] .
E ϕ ( s ) = 1 r sin θ 2 ( r Π ( s ) e ) r ϕ k 2 ( I ) r ( r Π ( s ) m ) θ = sin ϕ k ( I ) r l = 0 [ k 2 ( I ) k ( I ) B l y ζ l ( 1 ) ( k ( I ) r ) P l ( 1 ) ( cos θ ) sin θ B l x ζ l ( 1 ) ( k ( I ) r ) P l ( 1 ) ( cos θ ) 1 sin θ ] .
H r ( s ) = k 2 r Π ( s ) m + 2 ( r Π ( s ) m ) r 2 = 1 k ( I ) 2 r 2 l = 0 l ( l + 1 ) B l y ζ l ( 1 ) ( k ( I ) r ) P l ( 1 ) ( cos θ ) sin ϕ .
H θ ( s ) = k 1 ( I ) r sin θ ( r Π ( s ) e ) ϕ + 1 r 2 ( r Π ( s ) m ) r θ = sin ϕ k ( I ) l = 0 [ k 1 ( I ) k ( I ) B l x ζ l ( 1 ) ( k ( I ) r ) P l ( 1 ) ( cos θ ) 1 r sin θ B l y ζ l ( 1 ) ( k ( I ) r ) P l ( 1 ) ( cos θ ) sin θ r ] .
H ϕ ( s ) = k 1 ( I ) r ( r Π ( s ) e ) θ + 1 r sin θ 2 ( r Π ( s ) m ) r ϕ = 1 k ( I ) r l = 0 [ B l y ζ l ( 1 ) ( k ( I ) r ) P l ( 1 ) ( cos θ ) cos ϕ sin θ [ B l x P l ( 1 ) ( cos θ ) cos ϕ + B l z P l ( cos θ ) ] ζ l ( 1 ) ( k ( I ) r ) k 1 ( I ) sin θ k ( I ) ] .
sin Ω 0 = λ e n s ( 2 P 0 N 0 ) ,
A ( θ ) = A 2 λ sin θ [ D ext J 1 ( π D ext λ sin θ ) D int J 1 ( π D int λ sin θ ) ] ,
A ( x ) = i = 0 N 1 k i + 1 x [ α i + 1 J 1 ( α i + 1 x ) α i J 1 ( α i x ) ] .
A ( x ) = i = 0 n 1 γ i + 1 x [ α i + 1 J 1 ( α i + 1 x ) α i J 1 ( α i x ) ] .

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