Abstract

Using an analytical expression for an integral involving Bessel and Legendre functions, we succeed in obtaining the partial wave decomposition of a general optical beam at an arbitrary location relative to the origin. We also showed that solid angle integration will eliminate the radial dependence of the expansion coefficients. The beam shape coefficients obtained are given by an exact expression in terms of single or double integrals. These integrals can be evaluated numerically on a short time scale. We present the results for the case of a linear-polarized Gaussian beam.

© 2006 Optical Society of America

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References

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  2. J. D. Jackson, Classical Electrodynamics (Wiley, 1999).
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    [Crossref]
  5. G. Gouesbet, B. Maheu, and G. Grehan, J. Opt. Soc. Am. A 5, 1427 (1988).
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  6. G. Gouesbet, C. Letellier, K. F. Ren, and G. Grehan, Appl. Opt. 35, 1537 (1996).
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  7. J. A. Lock and G. Gouesbet, J. Opt. Soc. Am. A 11, 2503 (1994).
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  8. G. Gouesbet and J. A. Lock, J. Opt. Soc. Am. A 11, 2516 (1994).
    [Crossref]
  9. G. Gouesbet, G. Grehan, and B Maheu, Appl. Opt. 27, 4874 (1988).
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  10. G. Gouesbet, J. A. Lock, and G. Grehan, Appl. Opt. 34, 2133 (1995).
    [Crossref] [PubMed]
  11. A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, J. Phys. A 39, L293 (2006).
    [Crossref]
  12. L. W. Davis, Phys. Rev. A 19, 1177 (1979).
    [Crossref]
  13. J. P. Barton and D. R. Alexander, J. Appl. Phys. 66, 2800 (1989).
    [Crossref]
  14. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006).
  15. B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
    [Crossref]

2006 (1)

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, J. Phys. A 39, L293 (2006).
[Crossref]

1998 (1)

1996 (1)

1995 (1)

1994 (2)

1989 (1)

J. P. Barton and D. R. Alexander, J. Appl. Phys. 66, 2800 (1989).
[Crossref]

1988 (2)

1982 (1)

G. Gouesbet and G. Grehan, J. Opt. 13, 97 (1982).
[Crossref]

1979 (1)

L. W. Davis, Phys. Rev. A 19, 1177 (1979).
[Crossref]

1959 (1)

B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
[Crossref]

Alexander, D. R.

J. P. Barton and D. R. Alexander, J. Appl. Phys. 66, 2800 (1989).
[Crossref]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, 1995).

Barbosa, L. C.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, J. Phys. A 39, L293 (2006).
[Crossref]

Barton, J. P.

J. P. Barton and D. R. Alexander, J. Appl. Phys. 66, 2800 (1989).
[Crossref]

Cesar, C. L.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, J. Phys. A 39, L293 (2006).
[Crossref]

Cruz, C. H. B.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, J. Phys. A 39, L293 (2006).
[Crossref]

Davis, L. W.

L. W. Davis, Phys. Rev. A 19, 1177 (1979).
[Crossref]

Fontes, A.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, J. Phys. A 39, L293 (2006).
[Crossref]

Gouesbet, G.

Grehan, G.

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

Letellier, C.

Lock, J. A.

Maheu, B

Maheu, B.

Neves, A. A. R.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, J. Phys. A 39, L293 (2006).
[Crossref]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006).

Padilha, L. A.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, J. Phys. A 39, L293 (2006).
[Crossref]

Ren, K. F.

Richards, B.

B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
[Crossref]

Rodriguez, E.

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, J. Phys. A 39, L293 (2006).
[Crossref]

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, 1995).

Wolf, E.

B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
[Crossref]

Appl. Opt. (4)

J. Appl. Phys. (1)

J. P. Barton and D. R. Alexander, J. Appl. Phys. 66, 2800 (1989).
[Crossref]

J. Opt. (1)

G. Gouesbet and G. Grehan, J. Opt. 13, 97 (1982).
[Crossref]

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

J. Phys. A (1)

A. A. R. Neves, L. A. Padilha, A. Fontes, E. Rodriguez, C. H. B. Cruz, L. C. Barbosa, and C. L. Cesar, J. Phys. A 39, L293 (2006).
[Crossref]

Phys. Rev. A (1)

L. W. Davis, Phys. Rev. A 19, 1177 (1979).
[Crossref]

Proc. R. Soc. London Ser. A (1)

B. Richards and E. Wolf, Proc. R. Soc. London Ser. A 253, 358 (1959).
[Crossref]

Other (3)

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge U. Press, 2006).

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, 1995).

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Figures (2)

Fig. 1
Fig. 1

Absolute radial electric field for an overfilled beam reconstructed from the partial wave.

Fig. 2
Fig. 2

Absolute radial electric field for an overfilled beam from ASR theory.

Equations (9)

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F inc = E 0 Z 0 , 1 n , m [ G n m TE , TM j n ( k r ) X n m ( θ , ϕ ) ± i k G n m TM , TE × j n ( k r ) X n m ( θ , ϕ ) ] ,
j n ( k r ) [ G n m TM G n m TE ] = k r E 0 n ( n + 1 ) Y n m * ( θ , ϕ ) [ E r Z H r ] d Ω .
0 π d θ sin θ P n m ( cos θ ) exp ( i k r cos α cos θ ) J m ( k r sin α sin θ ) = 2 i n m P n m ( cos α ) j n ( k r ) ,
F ( ρ , φ , z ) = i k f exp ( i k f ) 2 π 0 α max d α sin α exp ( i k z cos α ) × 0 2 π d β F ( α , β ) exp [ i k ρ sin α cos ( β φ ) ] ,
F ( θ , ϕ ) = cos θ { [ ( F x F y 0 ) ( sin ϕ cos ϕ 0 ) ] ( sin ϕ cos ϕ 0 ) + [ ( F x F y 0 ) ( cos ϕ sin ϕ 0 ) ] ( cos ϕ cos θ sin ϕ cos θ sin θ ) } ,
[ G n m TM G n m TE ] = ± i k f exp ( i k f ) i n E 0 2 n + 1 4 π n ( n + 1 ) ( n m ) ! ( n + m ) ! × 0 α max d α sin α cos α exp ( i k z o cos α ) × 0 2 π d β exp [ i k ρ o sin α cos ( ϕ 0 β ) ] exp ( i m β ) × [ ( F x sin β F y cos β ) m sin α P n m ( cos α ) + i ( F x cos β + F y sin β ) d d α P n m ( cos α ) ]
[ G n m TM G n m TE ] = i k f exp ( i k f ) 2 π i n m exp ( i m ϕ o ) E 0 2 n + 1 4 π n ( n + 1 ) ( n m ) ! ( n + m ) ! 0 α max d α cos α exp ( i k z o cos α ) × { [ m 2 J m ( k ρ o sin α ) k ρ o sin α P n m ( cos α ) sin 2 α J m ( k ρ o sin α ) P n m ( cos α ) ] ( F x cos ϕ o + F y sin ϕ o ) + i m [ m J m ( k ρ o sin α ) P n m ( cos α ) sin 2 α J m ( k ρ o sin α ) k ρ o sin α P n m ( cos α ) ] ( F y cos ϕ o F x sin ϕ o ) } .
[ G n m TM G n m TE ] = ± 2 π i k f exp ( i k f ) i n m exp ( i m ϕ o ) 2 n + 1 4 π n ( n + 1 ) ( n m ) ! ( n + m ) ! 0 α max d α cos α exp ( f 2 sin 2 α ω a 2 ) exp ( i k z o cos α ) { [ m 2 J m ( k ρ o sin α ) k ρ o sin α × P n m ( cos α ) sin 2 α J m ( k ρ o sin α ) P n m ( cos α ) ] cos ϕ o + i m [ m J m ( k ρ o sin α ) P n m ( cos α ) sin 2 α J m ( k ρ o sin α ) k ρ o sin α P n m ( cos α ) ] sin ϕ o } .
[ G n , 1 TM G n , 1 TE ] = ± G n = ± π i k f exp ( i k f ) i n 1 n ( n + 1 ) × 2 n + 1 4 π 0 α max d α cos α exp ( f 2 sin 2 α ω a 2 ) × exp ( i k z o cos α ) [ P n 1 ( cos α ) sin 2 α P n 1 ( cos α ) ] ,

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