Abstract

Analytic and numerical results are given for the elastic scattering of evanescent electromagnetic waves by dielectric spheres. Some polarization and symmetry effects not found in Lorenz-Mie scattering are noted. The possibility of experimental studies is also discussed.

© 1979 Optical Society of America

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References

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  1. N. J. Harrick, Internal Reflection Spectroscopy (Interscience-Wiley, New York, 1967).
  2. T. Hirschfeld, M. J. Block, W. Mueller, J. Histochem. Cytochem. 25, 719 (1977).
    [CrossRef] [PubMed]
  3. E.-H. Lee, R. E. Benner, J. B. Fenn, R. K. Chang, Appl. Opt. 18, 862 (1979).
    [CrossRef]
  4. G. J. Rosasco, E. S. Etz, W. A. Cassatt, Appl. Spectrosc. 29, 396 (1975).
    [CrossRef]
  5. H. Chew, P. J. McNulty, M. Kerker, Phys. Rev. A13, 396 (1976); H. Chew, M. Kerker, P. J. McNulty, J. Opt. Soc. Am. 66, 440 (1976); M. Kerker, P. J. McNulty, M. Sculley, H. Chew, D. D. Cooke, J. Opt. Soc. Am. 68, 1676 (1978); H. Chew, M. Sculley, M. Kerker, P. J. McNulty, D. D. Cooke, J. Opt. Soc. Am. 68, 1686 (1978); J. P. Kratohvil, M.-P. Lee, M. Kerker, Appl. Opt. 17, 1978 (1978); M. Kerker, S. D. Druger, Appl. Opt. 18, 1172 (1979); P. J. McNulty, S. D. Druger, M. Kerker, H. Chew, Appl. Opt. 18, 1484 (1979).
    [CrossRef] [PubMed]
  6. A. Ashkin, J. Dziedzic, Phys. Rev. Lett. 36, 367 (1976); Appl. Phys. Lett. 28, 333 (1976); Appl. Phys. Lett. 30, 202 (1977).
    [CrossRef]
  7. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  8. See, for example, J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975). Our x,y,z axes correspond to the z,-y,x axes of this reference.
  9. We use the same notations as Ref. 5, Chew, McNulty, Kerker (1976).
  10. See, for example, Appendix B of Ref. 5, Chew, McNulty, Kerker (1976).
  11. The fields and their expansions in Eqs. (8), (10), (11), (18)–(20) refer to the new origin, i.e., the center of the dielectric sphere.

1979 (1)

1977 (1)

T. Hirschfeld, M. J. Block, W. Mueller, J. Histochem. Cytochem. 25, 719 (1977).
[CrossRef] [PubMed]

1976 (2)

H. Chew, P. J. McNulty, M. Kerker, Phys. Rev. A13, 396 (1976); H. Chew, M. Kerker, P. J. McNulty, J. Opt. Soc. Am. 66, 440 (1976); M. Kerker, P. J. McNulty, M. Sculley, H. Chew, D. D. Cooke, J. Opt. Soc. Am. 68, 1676 (1978); H. Chew, M. Sculley, M. Kerker, P. J. McNulty, D. D. Cooke, J. Opt. Soc. Am. 68, 1686 (1978); J. P. Kratohvil, M.-P. Lee, M. Kerker, Appl. Opt. 17, 1978 (1978); M. Kerker, S. D. Druger, Appl. Opt. 18, 1172 (1979); P. J. McNulty, S. D. Druger, M. Kerker, H. Chew, Appl. Opt. 18, 1484 (1979).
[CrossRef] [PubMed]

A. Ashkin, J. Dziedzic, Phys. Rev. Lett. 36, 367 (1976); Appl. Phys. Lett. 28, 333 (1976); Appl. Phys. Lett. 30, 202 (1977).
[CrossRef]

1975 (1)

Ashkin, A.

A. Ashkin, J. Dziedzic, Phys. Rev. Lett. 36, 367 (1976); Appl. Phys. Lett. 28, 333 (1976); Appl. Phys. Lett. 30, 202 (1977).
[CrossRef]

Benner, R. E.

Block, M. J.

T. Hirschfeld, M. J. Block, W. Mueller, J. Histochem. Cytochem. 25, 719 (1977).
[CrossRef] [PubMed]

Cassatt, W. A.

Chang, R. K.

Chew,

We use the same notations as Ref. 5, Chew, McNulty, Kerker (1976).

See, for example, Appendix B of Ref. 5, Chew, McNulty, Kerker (1976).

Chew, H.

H. Chew, P. J. McNulty, M. Kerker, Phys. Rev. A13, 396 (1976); H. Chew, M. Kerker, P. J. McNulty, J. Opt. Soc. Am. 66, 440 (1976); M. Kerker, P. J. McNulty, M. Sculley, H. Chew, D. D. Cooke, J. Opt. Soc. Am. 68, 1676 (1978); H. Chew, M. Sculley, M. Kerker, P. J. McNulty, D. D. Cooke, J. Opt. Soc. Am. 68, 1686 (1978); J. P. Kratohvil, M.-P. Lee, M. Kerker, Appl. Opt. 17, 1978 (1978); M. Kerker, S. D. Druger, Appl. Opt. 18, 1172 (1979); P. J. McNulty, S. D. Druger, M. Kerker, H. Chew, Appl. Opt. 18, 1484 (1979).
[CrossRef] [PubMed]

Dziedzic, J.

A. Ashkin, J. Dziedzic, Phys. Rev. Lett. 36, 367 (1976); Appl. Phys. Lett. 28, 333 (1976); Appl. Phys. Lett. 30, 202 (1977).
[CrossRef]

Etz, E. S.

Fenn, J. B.

Harrick, N. J.

N. J. Harrick, Internal Reflection Spectroscopy (Interscience-Wiley, New York, 1967).

Hirschfeld, T.

T. Hirschfeld, M. J. Block, W. Mueller, J. Histochem. Cytochem. 25, 719 (1977).
[CrossRef] [PubMed]

Jackson, J. D.

See, for example, J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975). Our x,y,z axes correspond to the z,-y,x axes of this reference.

Kerker,

We use the same notations as Ref. 5, Chew, McNulty, Kerker (1976).

See, for example, Appendix B of Ref. 5, Chew, McNulty, Kerker (1976).

Kerker, M.

H. Chew, P. J. McNulty, M. Kerker, Phys. Rev. A13, 396 (1976); H. Chew, M. Kerker, P. J. McNulty, J. Opt. Soc. Am. 66, 440 (1976); M. Kerker, P. J. McNulty, M. Sculley, H. Chew, D. D. Cooke, J. Opt. Soc. Am. 68, 1676 (1978); H. Chew, M. Sculley, M. Kerker, P. J. McNulty, D. D. Cooke, J. Opt. Soc. Am. 68, 1686 (1978); J. P. Kratohvil, M.-P. Lee, M. Kerker, Appl. Opt. 17, 1978 (1978); M. Kerker, S. D. Druger, Appl. Opt. 18, 1172 (1979); P. J. McNulty, S. D. Druger, M. Kerker, H. Chew, Appl. Opt. 18, 1484 (1979).
[CrossRef] [PubMed]

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

Lee, E.-H.

McNulty,

See, for example, Appendix B of Ref. 5, Chew, McNulty, Kerker (1976).

We use the same notations as Ref. 5, Chew, McNulty, Kerker (1976).

McNulty, P. J.

H. Chew, P. J. McNulty, M. Kerker, Phys. Rev. A13, 396 (1976); H. Chew, M. Kerker, P. J. McNulty, J. Opt. Soc. Am. 66, 440 (1976); M. Kerker, P. J. McNulty, M. Sculley, H. Chew, D. D. Cooke, J. Opt. Soc. Am. 68, 1676 (1978); H. Chew, M. Sculley, M. Kerker, P. J. McNulty, D. D. Cooke, J. Opt. Soc. Am. 68, 1686 (1978); J. P. Kratohvil, M.-P. Lee, M. Kerker, Appl. Opt. 17, 1978 (1978); M. Kerker, S. D. Druger, Appl. Opt. 18, 1172 (1979); P. J. McNulty, S. D. Druger, M. Kerker, H. Chew, Appl. Opt. 18, 1484 (1979).
[CrossRef] [PubMed]

Mueller, W.

T. Hirschfeld, M. J. Block, W. Mueller, J. Histochem. Cytochem. 25, 719 (1977).
[CrossRef] [PubMed]

Rosasco, G. J.

Appl. Opt. (1)

Appl. Spectrosc. (1)

J. Histochem. Cytochem. (1)

T. Hirschfeld, M. J. Block, W. Mueller, J. Histochem. Cytochem. 25, 719 (1977).
[CrossRef] [PubMed]

Phys. Rev. (1)

H. Chew, P. J. McNulty, M. Kerker, Phys. Rev. A13, 396 (1976); H. Chew, M. Kerker, P. J. McNulty, J. Opt. Soc. Am. 66, 440 (1976); M. Kerker, P. J. McNulty, M. Sculley, H. Chew, D. D. Cooke, J. Opt. Soc. Am. 68, 1676 (1978); H. Chew, M. Sculley, M. Kerker, P. J. McNulty, D. D. Cooke, J. Opt. Soc. Am. 68, 1686 (1978); J. P. Kratohvil, M.-P. Lee, M. Kerker, Appl. Opt. 17, 1978 (1978); M. Kerker, S. D. Druger, Appl. Opt. 18, 1172 (1979); P. J. McNulty, S. D. Druger, M. Kerker, H. Chew, Appl. Opt. 18, 1484 (1979).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

A. Ashkin, J. Dziedzic, Phys. Rev. Lett. 36, 367 (1976); Appl. Phys. Lett. 28, 333 (1976); Appl. Phys. Lett. 30, 202 (1977).
[CrossRef]

Other (6)

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

See, for example, J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975). Our x,y,z axes correspond to the z,-y,x axes of this reference.

We use the same notations as Ref. 5, Chew, McNulty, Kerker (1976).

See, for example, Appendix B of Ref. 5, Chew, McNulty, Kerker (1976).

The fields and their expansions in Eqs. (8), (10), (11), (18)–(20) refer to the new origin, i.e., the center of the dielectric sphere.

N. J. Harrick, Internal Reflection Spectroscopy (Interscience-Wiley, New York, 1967).

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Figures (10)

Fig. 1
Fig. 1

Propagation of wave k at incident angle I from dense medium μ,,n into medium μ′,′,n′ with propagation vector k′ and refraction angle r. When I > I0 = sin−1(n′/n), only an evanescent wave is present.

Fig. 2
Fig. 2

Particle with radius a and μl, 1, and n1 located at x = d in medium n′.

Fig. 3
Fig. 3

Differential scattering cross section σ11 vs scattering angle in the xz′ plane for a single sphere with a = 460 nm, k′ = 0.0184 nm−1, d =2a, n1/n′ = 1.112, and critical angle I0 = 54.3°. The incident and scattered wave are polarized perpendicular to the incident plane. Values of the incident angle are I = 54.3°, 54.4°, 55°, and 60°.

Fig. 4
Fig. 4

Same as Fig. 3 with the incident and scattered waves polarized parallel to the incident plane. Differential scattering cross section is σ22.

Fig. 5
Fig. 5

Differential scattering cross sections σ11 and σ12 vs scattering angle in the yz′ plane for I = 54.3° (labeled Mie, σ12 = 0) and for I = 60°. Other parameters are the same as in Fig. 3. The incident wave is polarized perpendicular to the incident plane.

Fig. 6
Fig. 6

Same as Fig. 5 with the incident wave polarized parallel to the incident plane. Differential scattering cross sections are σ22 and σ21. σ21 = 0 for I = 54.3°.

Fig. 7
Fig. 7

Same as Fig. 3 for n1/n′ = 1.33.

Fig. 8
Fig. 8

Same as Fig. 4 for n1/n′ = 1.33.

Fig. 9
Fig. 9

Same as Fig. 5 for n1/n′ = 1.33.

Fig. 10
Fig. 10

Same as Fig. 6 for n1/n′ = 1.33.

Equations (37)

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E = ^ 2 exp ( i k · r ) ,
E = E 0 ^ 2 exp ( i k · r ) ,
E inc = E 0 ^ 2 exp ( i k · r ) = l , m { i c n 2 ω α E ( l , m ) × [ j l ( k r ) Y l l m ( r ^ ) ] + α M ( l , m ) j l ( k r ) Y l l m ( r ^ ) } ,
B inc = l , m { α E ( l , m ) j l ( k r ) Y l l m ( r ^ ) - i c ω α M ( l , m ) × [ j l ( k r ) Y l l m ( r ) ] } .
α M ( l , m ) j l ( k r ) = Y l l m * ( r ^ ) · E inc d Ω r = E 0 [ l ( l + 1 ) ] 1 / 2 [ L y Y l m ( r ^ ) ] * exp ( i k · r ) d Ω r = 2 π i l + 1 [ l ( l + 1 ) ] 1 / 2 j l ( k r ) { [ ( l - m ) ( l + m + 1 ) ] 1 / 2 Y l , m + 1 * ( k ^ ) - [ ( l + m ) ( l - m + 1 ) ] 1 / 2 Y l , m - 1 * ( k ^ ) } ,
α M ( l , m ) = i l + 1 [ π ( 2 l + 1 ) ( l - m ) ! l ( l + 1 ) ( l + m ) ! ] 1 / 2 [ P l m + 1 ( cos θ k ) - ( l + m ) ( l - m + 1 ) P l m - 1 ( cos θ k ) ] * E 0 , = - 2 i l + 1 [ π ( 2 l + 1 ) ( l - m ) ! l ( l + 1 ) ( l + m ) ! ] 1 / 2 [ d d θ k p l m ( cos θ k ) ] * E 0 }
k i n ( 2 l + 1 ) [ l j l + 1 2 ( k r ) + ( l + 1 ) j l - 1 2 ( k r ) ] α E ( l , m ) = { × [ j l ( k r ) Y l l m ( r ^ ) ] } * · E inc d Ω = 2 π k i l - 1 [ l ( l + 1 ) ( 2 l + 1 ) ] 1 / 2 E 0 [ ( l + 1 ) j l - 1 2 ( k r ) + l j l + 1 2 ( k r ) ] · [ l ( l - m ) ! 4 π ( l + m ) ! ] 1 / 2 [ ( l + m ) ( l + m - 1 ) P l - 1 m - 1 ( cos θ k ) + P l - 1 m + 1 ( cos θ k ) ] * ,
α E ( l , m ) = i l - 2 n [ π ( l - m ) ! ( 2 l + 1 ) l ( l + 1 ) ( l + m ) ! ] 1 / 2 [ ( l + m ) ( l + m - 1 ) P l - 1 m - 1 ( cos θ k ) + P l - 1 m + 1 ( cos θ k ) ] * E 0 = 2 m i l n [ π ( 2 l + 1 ) ( l - m ) ! l ( l + 1 ) ( l + m ) ! ] 1 / 2 [ P l m ( cos θ k ) / sin θ k ] * E 0 ,
E inc E 0 ^ 2 exp [ i k z ( n / n ) sin I - k ( n 2 sin 2 I / n 2 - 1 ) 1 / 2 x ] E 0 ^ 2 exp ( i α z - β x )
E s c ( r ) = l , m { i c n 2 ω β E ( l , m ) × [ h l ( 1 ) ( k r ) Y l l m ( r ^ ) ] + β M ( l , m ) h l ( 1 ) ( k r ) Y l l m ( r ^ ) } large r exp ( i k r ) k r l , m ( - i ) l - 1 { β E ( l , m ) n r ^ × Y l l m ( r ^ ) - β M ( l , m ) Y l l m ( r ^ ) } ,
β E ( l , m ) = { j l ( k a ) [ k 1 a j l ( k 1 a ) ] - 1 j l ( k 1 a ) [ k a j l ( k a ) ] } exp ( - β d ) α E ( l , m ) 1 j l ( k 1 a ) [ k a h l ( 1 ) ( k a ) ] - h l ( 1 ) ( k a ) [ k 1 a j l ( k 1 a ) ] ,
β M ( l , m ) = { μ 1 j l ( k 1 a ) [ k a j l ( k a ) ] - μ j l ( k a ) [ k 1 a j l ( k 1 a ) ] } exp ( - β d ) α M ( l , m ) μ h l ( 1 ) ( k a ) [ k 1 a j l ( k 1 a ) ] - μ 1 j l ( k 1 a ) [ k a h l ( 1 ) ( k a ) ] ,
E int = l , m { i c n 1 2 ω γ E ( l , m ) × [ j l ( k 1 r ) Y l l m ( r ^ ) ] + γ M ( l , m ) j l ( k 1 r ) Y l l m ( r ^ ) } ,
B int = l , m { γ E ( l , m ) j l ( k 1 r ) Y l l m ( r ^ ) - ( i c ω ) γ M ( l , m ) × [ j l ( k 1 r ) Y l l m ( r ^ ) ] } ,
γ E ( l , m ) = ( i n 1 2 / μ k a ) exp ( - β d ) α E ( l , m ) 1 j l ( k 1 a ) [ k a h l ( 1 ) ( k a ) ] - h l ( 1 ) ( k a ) [ k 1 a j l ( k 1 a ) ] .
γ M ( l , m ) = ( i μ 1 / k a ) exp ( - β d ) α M ( l , m ) μ 1 j l ( k 1 a ) [ k a h l ( 1 ) ( k a ) ] - μ h l ( 1 ) ( k a ) [ k 1 a j l ( k 1 a ) ] .
E = ( sin I , 0 , - cos I ) exp ( i k · r ) ,
E ˜ = E ˜ 0 ( sin r , 0 , - cos r ) exp ( i k · r )
α ˜ M ( l , m ) j l ( k r ) = Y l l m * ( r ^ ) · E ˜ d Ω r , = 4 π i l j l ( k r ) [ l ( l + 1 ) ] 1 / 2 ( sin r 2 { [ ( l - m ) ( l + m + 1 ) ] 1 / 2 Y l , m + 1 * ( k ^ ) + [ ( l + m ) ( l - m + 1 ) ] 1 / 2 Y l , m - 1 * ( k ^ ) } - m cos r Y l m * ( k ^ ) ) * E ˜ 0
i ( ω / c ) 2 l + 1 [ l j l + 1 2 ( k r ) + ( l + 1 ) j l - 1 2 ( k r ) ] α ˜ E ( l , m ) = { × [ j l ( k r ) Y l l m ( r ^ ) ] } * · E ˜ d Ω r = i l k [ π ( l - m ) ! l ( l + 1 ) ( 2 l + 1 ) ( l + m ) ! ] 1 / 2 [ ( l + 1 ) j l - 1 2 ( k r ) + l j l + 1 2 ( k r ) ] { sin r [ ( l + m ) ( l + m - 1 ) P l - 1 m - 1 ( cos θ k ) - P l - 1 m + 1 ( cos θ k ) ] + 2 ( l + m ) cos r P l - 1 m ( cos θ k ) } * E ˜ 0 .
α ˜ M ( l , m ) = [ π ( 2 l + 1 ) ( l - m ) ! l ( l + 1 ) ( l + m ) ! ] 1 / 2 i l { cos θ k [ P l m + 1 ( cos θ k ) + ( l + m ) ( l - m + 1 ) P l m - 1 ( cos θ k ) ] - 2 m sin θ k P l m ( cos θ k ) } * E ˜ 0 = - 2 m i l [ π ( 2 l + 1 ) ( l - m ) ! l ( l + 1 ) ( l + m ) ! ] 1 / 2 [ P l m ( cos θ k ) / sin θ k ] * E ˜ 0 ,
α ˜ E ( l , m ) = i l - 1 [ ( 2 l + 1 ) π ( l - m ) ! l ( l + 1 ) ( l + m ) ! ] 1 / 2 { cos θ k [ ( l + m ) ( l + m - 1 ) P l - 1 m + 1 ( cos θ k ) - P l - 1 m + 1 ( cos θ k ) ] + 2 ( l + m ) sin θ k P l - 1 m ( cos θ k ) } * E ˜ 0
= - 2 i l - 1 n [ π ( 2 l + 1 ) ( l - m ) ! l ( l + 1 ) ( l + m ) ! ] 1 / 2 [ d d θ k P l m ( cos θ k ) ] * E ˜ 0 .
E s c ( r ) = l , m { i c n 2 ω β ˜ E ( l , m ) × [ h l ( 1 ) ( k r ) Y l l m ( r ^ ) ] + β ˜ M ( l , m ) h l ( 1 ) ( k r ) Y l l m ( r ^ ) } large r exp ( i k r ) k r l , m ( - i ) l - 1 [ 1 n β ˜ E ( l , m ) r ^ × Y l l m ( r ^ ) - β ˜ M ( l , m ) Y l l m ( r ^ ) ] ,
E int ( r ) = l , m { i c n 1 2 ω γ ˜ E ( l , m ) × [ j l ( k 1 r ) Y l l m ( r ^ ) ] + γ M ( l , m ) j l ( k 1 r ) Y l l m ( r ) } ,
B int ( r ) = l , m { γ ˜ E ( l , m ) j l ( k 1 r ) Y l l m ( r ^ ) - ( i c / ω ) γ ˜ M ( l , m ) × [ j l ( k 1 r ) Y l l m ( r ^ ) ] } ,
β ˜ E , M ( l , m ) = the right - hand side of Eqs . ( 9 a ) and ( 9 b ) with α E , M ( l , m ) replaced by α ¯ E , M ( l , m ) ,
γ ˜ E , M ( l , m ) = the right - hand side of Eqs . ( 12 a ) and ( 12 b ) with α E , M ( l , m ) replaced by α ¯ E , M ( l , m ) .
Y l l m ( θ , 0 ) = - N l m [ m sin θ P l m ( cos θ ) θ ^ + i d d θ P l m ( cos θ ) ϕ ^ ] , [ r ^ × Y l l m ( θ , ϕ ) ] ϕ = 0 = N l m [ i d d θ P l m ( cos θ ) θ ^ - m sin θ P l m ( cos θ ) ϕ ^ ] ,
E s c ( r ) large r exp ( i k r ) k r l , m ( - i ) l - 1 [ 1 n β E ( l , m ) r ^ × Y l l m ( θ , ϕ ) - β M ( l , m ) Y l l m ( θ , ϕ ) ] ϕ = 0 ,
β E ( l , m ) = a l exp ( - β d ) α E ( l , m ) = 4 π exp ( - β d ) E 0 n i l a l N l m m [ P l m ( cos θ k ) sin θ k ] * ,
β M ( l , m ) = 4 π exp ( - β d ) E 0 i l + 1 b l N l m [ d d θ k P l m ( cos θ k ) ] * ,
- l a l m = - l m = l m N l m 2 [ P l m ( cos θ k ) sin θ k d P l m ( cos θ ) d θ ] * + l b l m = - l m = l m N l m 2 [ d P l m ( cos θ k ) d θ k · P l m ( cos θ ) sin θ ] * .
P l - m = ( l - m ) ! ( l + m ) ! ( - 1 ) m P l m .
E 0 2 = 4 1 + tan I / tan r 2 = 3.87
E 0 2 = 16 sin 2 r cos 2 I sin 2 r + sin 2 I 2 = 5.63
E 0 2 exp ( - 2 β d ) 0.05 ( polarization plane of incidence ) , 0.073 ( polarization in plane of incidence ) .

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