Abstract

The closed form of the complete far-field asymptotic series is provided for the scalar and electromagnetic monochromatic fields in free space whose angular spectrum is smooth over the sphere of directions. The first few terms of this series are also shown to give corrections to the Fraunhofer diffraction formula. The application of this series is illustrated with two examples corresponding to a highly focused radially polarized field and to the scalar diffraction from a circular aperture.

© 2006 Optical Society of America

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2006

2004

2003

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

2001

1999

1983

1982

1976

G. C. Sherman, J. J. Stamnes, and E. Lalor, J. Math. Phys. 17, 760 (1976).
[CrossRef]

1973

A. J. Devaney and G. C. Sherman, SIAM Rev. 15, 765 (1973).
[CrossRef]

1952

D. S. Jones, Proc. Cambridge Philos. Soc. 48, 733 (1952).
[CrossRef]

1909

P. Debye, Ann. Phys. 30, 755 (1909).
[CrossRef]

1902

E. T. Whittaker, Math. Ann. 57, 333 (1902).
[CrossRef]

Alonso, M. A.

Biss, D. P.

Bokor, N.

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 484-498.

Brown, T. G.

Chew, W. C.

Davidson, N.

Debye, P.

P. Debye, Ann. Phys. 30, 755 (1909).
[CrossRef]

Devaney, A. J.

A. J. Devaney and G. C. Sherman, SIAM Rev. 15, 765 (1973).
[CrossRef]

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Jones, D. S.

D. S. Jones, Proc. Cambridge Philos. Soc. 48, 733 (1952).
[CrossRef]

Lalor, E.

G. C. Sherman, J. J. Stamnes, and E. Lalor, J. Math. Phys. 17, 760 (1976).
[CrossRef]

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 128-141.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Saghafi, S.

Santarsiero, M.

Sheppard, C. J. R.

Sherman, G. C.

G. C. Sherman and W. C. Chew, J. Opt. Soc. Am. 72, 1076 (1982).
[CrossRef]

G. C. Sherman, J. J. Stamnes, and E. Lalor, J. Math. Phys. 17, 760 (1976).
[CrossRef]

A. J. Devaney and G. C. Sherman, SIAM Rev. 15, 765 (1973).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, J. Opt. Soc. Am. 73, 96 (1983).
[CrossRef]

G. C. Sherman, J. J. Stamnes, and E. Lalor, J. Math. Phys. 17, 760 (1976).
[CrossRef]

Whittaker, E. T.

E. T. Whittaker, Math. Ann. 57, 333 (1902).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 484-498.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 128-141.

Ann. Phys.

P. Debye, Ann. Phys. 30, 755 (1909).
[CrossRef]

J. Math. Phys.

G. C. Sherman, J. J. Stamnes, and E. Lalor, J. Math. Phys. 17, 760 (1976).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Math. Ann.

E. T. Whittaker, Math. Ann. 57, 333 (1902).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef] [PubMed]

Proc. Cambridge Philos. Soc.

D. S. Jones, Proc. Cambridge Philos. Soc. 48, 733 (1952).
[CrossRef]

SIAM Rev.

A. J. Devaney and G. C. Sherman, SIAM Rev. 15, 765 (1973).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), p. 484-498.

Ref. , pp. 921-923.

Ref. , pp. 436-475.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), pp. 128-141.

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Figures (2)

Fig. 1
Fig. 1

Rms error as a function of radius for the approximations to the field in Eq. (11) with k q = 1 , resulting from the truncation of the asymptotic series to zeroth, first, second, and third order.

Fig. 2
Fig. 2

Relative error ν N as a function of the polar angle θ, at a distance of r = 50 λ , of the estimates in Eq. (15) for a scalar plane wave through a circular aperture whose radius is 5 λ . The black curve gives the shape of the modulus of the exact field.

Equations (15)

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U ( r ) = 1 4 π 4 π A ( u ̂ ) exp ( i k u ̂ r ) d Ω ,
U ( r ) A ( r ̂ ) exp ( i k r ) 2 i k r A ( r ̂ ) exp ( i k r ) 2 i k r ,
U ( r ) = n = 0 A n + ( r ̂ ) exp ( i k r ) A n ( r ̂ ) exp ( i k r ) 2 ( i k r ) n + 1 ,
( 1 r 2 r r 2 r + 1 r 2 Ω 2 + k 2 ) U ( r ) = 0 ,
Ω 2 = 1 sin θ θ sin θ θ + 1 sin 2 θ 2 ϕ 2 .
( 1 r 2 r r 2 r ) exp ( ± i k r ) r n + 1 = [ n ( n + 1 ) r 2 2 i k n r k 2 ] exp ( ± i k r ) r n + 1 ,
A n ± ( r ̂ ) = ± [ n ( n 1 ) + Ω 2 ] A n 1 ± ( r ̂ ) 2 n ,
A n ± ( r ̂ ) = ( ± 1 ) n 2 n n ! { j = 0 n 1 [ j ( j + 1 ) + Ω 2 ] } A ( r ̂ ) .
E ( r ) = 1 4 π 4 π A ( u ̂ ) exp ( i k u ̂ r ) d Ω ,
A ( u ̂ ) = E 0 ( z ̂ u z u ̂ ) exp ( k q u z ) ,
E ( r ) = E 0 ( 2 x z , 2 y z , k 2 + 2 z 2 ) sin ( k R ) k 3 R ,
ϵ N ( r ) = 4 π E ( r u ̂ ) E N ( r u ̂ ) 2 d Ω 4 π E ( r u ̂ ) 2 d Ω ,
E N ( r ) = n = 0 N 1 ( 2 i k r ) n + 1 n ! { j = 0 n 1 [ j ( j + 1 ) + Ω 2 ] } × [ A ( r ̂ ) exp ( i k r ) + ( 1 ) n A ( r ̂ ) exp ( i k r ) ] .
A ( u ) = 2 k a u z J 1 ( k a u ) u ,
U N ( r ) = n = 0 N 1 ( 2 i k r ) n n ! { j = 0 n 1 [ j ( j + 1 ) + Ω 2 ] } U FR ( r ) ,

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