Abstract

A scheme for computing rotationally-symmetric nonparaxial monochromatic scalar fields is proposed, based on a new orthonormal basis of solutions of the Helmholtz equation given by combinations of spherical waves focused at imaginary points. These basis fields are found through a mapping of the angular spectra of the multipolar basis over the sphere of directions. The convergence of the basis can be optimized by an appropriate choice of the location of the imaginary focus. The new scheme is tested for the case of converging spherical waves of different numerical apertures, with and without aberrations.

© 2006 Optical Society of America

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References

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  1. E. Heyman and L. B. Felsen, "Complex source pulse-beam fields," J. Opt. Soc. Am. A 6, 806-817 (1989).
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    [CrossRef]
  3. C. J. R. Sheppard, "High-aperture beams," J. Opt. Soc. Am. A 18, 1579-1587 (2001).
    [CrossRef]
  4. M. A. Alonso, R. Borghi, and M. Santarsiero, "Joint spatial-directional localization features of wave fields focused at a complex point," J. Opt. Soc. Am. A 23, 933-939 (2006).
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    [CrossRef]
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    [CrossRef]
  7. J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
    [CrossRef]
  8. D. Ding and Y. Zhang, "Notes on the Gaussian beam expansion," J. Acoust. Soc. Am. 116, 1401-1405 (2004).
    [CrossRef]
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    [CrossRef]
  10. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998), Chaps. 3 and 9.
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    [CrossRef]
  14. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999), Sec. 13.2.1.

2006

2004

D. Ding and Y. Zhang, "Notes on the Gaussian beam expansion," J. Acoust. Soc. Am. 116, 1401-1405 (2004).
[CrossRef]

2002

2001

2000

1998

C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: A scalar treatment," Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

1996

R. Borghi, F. Gori, and M. Santarsiero, "Optimization of Laguerre-Gauss truncated series," Opt. Commun. 125, 197-203 (1996).
[CrossRef]

1989

1988

B. Tehan Landesman and H. H. Barrett, "Gaussian amplitude functions that are exact solutions to the scalar Helmholtz equation," J. Opt. Soc. Am. A 5, 1610-1619 (1988).
[CrossRef]

J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

1974

1947

V. Bargmann, "Irreducible unitary representations of the Lorentz group," Ann. Math. 48, 568-640 (1947).
[CrossRef]

Alonso, M. A.

Bargmann, V.

V. Bargmann, "Irreducible unitary representations of the Lorentz group," Ann. Math. 48, 568-640 (1947).
[CrossRef]

Barrett, H. H.

Borghi, R.

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

Ding, D.

D. Ding and Y. Zhang, "Notes on the Gaussian beam expansion," J. Acoust. Soc. Am. 116, 1401-1405 (2004).
[CrossRef]

Felsen, L. B.

Gori, F.

R. Borghi, F. Gori, and M. Santarsiero, "Optimization of Laguerre-Gauss truncated series," Opt. Commun. 125, 197-203 (1996).
[CrossRef]

Heyman, E.

Li, Y.

Ludlow, I. K.

Saghafi, S.

C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: A scalar treatment," Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

Santarsiero, M.

Sheppard, C. J. R.

C. J. R. Sheppard, "High-aperture beams," J. Opt. Soc. Am. A 18, 1579-1587 (2001).
[CrossRef]

C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: A scalar treatment," Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

Siegman, A. E.

Sziklas, E. A.

Tehan Landesman, B.

Ulanowski, Z.

Wen, J. J.

J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

Zhang, Y.

D. Ding and Y. Zhang, "Notes on the Gaussian beam expansion," J. Acoust. Soc. Am. 116, 1401-1405 (2004).
[CrossRef]

Ann. Math.

V. Bargmann, "Irreducible unitary representations of the Lorentz group," Ann. Math. 48, 568-640 (1947).
[CrossRef]

Appl. Opt.

J. Acoust. Soc. Am.

J. J. Wen and M. A. Breazeale, "A diffraction beam field expressed as the superposition of Gaussian beams," J. Acoust. Soc. Am. 83, 1752-1756 (1988).
[CrossRef]

D. Ding and Y. Zhang, "Notes on the Gaussian beam expansion," J. Acoust. Soc. Am. 116, 1401-1405 (2004).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

R. Borghi, F. Gori, and M. Santarsiero, "Optimization of Laguerre-Gauss truncated series," Opt. Commun. 125, 197-203 (1996).
[CrossRef]

Opt. Lett.

Phys. Rev. A

C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: A scalar treatment," Phys. Rev. A 57, 2971-2979 (1998).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999), Sec. 13.2.1.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998), Chaps. 3 and 9.

Supplementary Material (6)

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» Media 4: MOV (140 KB)     
» Media 5: MOV (595 KB)     
» Media 6: MOV (566 KB)     

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Figures (9)

Fig. 1.
Fig. 1.

Mapping of the angular spectra over a meridional section of the sphere of directions. The lengths of the blue radial line segments are proportional to |A 0,0(u;q)|2. For increasing q, these radial lines bunch up around the positive z axis. [Media 1]

Fig. 2.
Fig. 2.

Plots of A l,0, for l = 0,1,2,3,4, as functions of θ/π, for q running from 0 to 4. [Media 2]

Fig. 3.
Fig. 3.

Plots of B l,0’s, as functions of τ, for several values of l.

Fig. 4.
Fig. 4.

Behavior of the rms error ε, as a function of a, for different truncation orders l max.

Fig. 5.
Fig. 5.

a) Angular spectrum A(θ) (green curve), and inverse-mapped spectrum A(θ) for the specified q (black curve) and for q = q opt (blue curve). b) Truncation error ε, for l max ranging from 1 to 30, corresponding to the multipolar expansion (orange dots), and to the expansion in the functions A l (black dots) for q running from 0 to 10. [Media 3]

Fig. 6.
Fig. 6.

a) Angular spectrum A(θ) (green curve), and inverse-mapped spectrum A(θ) for q = q opt (blue curve), for θ max running from 0 to π/2. b) Expansion coefficients cl for the multipolar basis (orange dots), and for the basis of functions Al (black dots). c) Truncation error ε, for l max ranging from 1 to 30, corresponding to the multipolar basis (orange dots), and to the basis of functions Al (black dots). [Media 4]

Fig. 7.
Fig. 7.

Contour plots of the field amplitude (normalized to the focal amplitude) at the focal region of an apertured spherical wave with half-angle θ max ∊ [0,π/2], calculated by using Eq. (31) with q = q opt and l max = 15. [Media 5]

Fig. 8.
Fig. 8.

a) Magnitude of the angular spectrum A(θ) in Eq. (32) (green curve), and inverse-mapped spectrum A (θ) for q = 3 (blue curve). b) Truncation errore, for l max ranging from 1 to 30, corresponding to the multipolar expansion (orange dots), and to the expansion in the functions Al (black dots) for q = 3, for the aberrated angular spectrum in Eq. (33) with s = 15.

Fig. 9.
Fig. 9.

Contour plots of the field amplitude (normalized to the unaberrated focal amplitude) at the focal region of the field corresponding to the angular spectrum in Eq. (33) for s between 0 and 30, calculated by using Eq. (31) with q = 3 and l max = 15. [Media 6]

Equations (33)

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U ( r ) = 4 π A ( u ) exp ( i k u · r ) ,
Y l , ± m ( u ) = ( ± 1 ) m 2 l + 1 4 π ( l m ) ! ( l + m ) ! exp ( ± i m ϕ ) P l ( m ) ( cos θ ) ,
4 π Y l , m ( u ) Y l , m * ( u ) = δ l , l δ m , m .
A ( u ) = l = 0 m = l + l a l , m Y l , m ( u ) ,
a l , m = 4 π Y l , m * ( u ) A ( u ) .
l , m ( r ) = 4 π Y l , m ( u ) exp ( iu · r ) = 4 π i l j l ( r ) Y l , m ( r r ) ,
A ( u ) = Y 0,0 ( u ) = 1 4 π , 0,0 ( r ) = 4 π sin r r .
U ( r ) U ( r r 0 ) A ( u ) A ( u ) exp ( iu · r 0 ) .
A 0,0 ( u ; q ) = ( q 2 π sinh 2 q ) 1 2 exp ( q cos θ ) ,
4 π Y 0,0 ( u ) 2 = 4 π A 0,0 ( u ; q ) 2 = 1 .
U 0,0 ( r ; q ) = ( 8 π q sinh 2 q ) 1 2 sin x 2 + y 2 + ( z i q ) 2 x 2 + y 2 + ( z i q ) 2 .
A 0,0 ( u ; q ) ( q 2 π ) 1 2 exp ( q θ 2 2 ) ,
U 0,0 ( r ; q ) 8 π q exp ( iz ) z i q exp [ i x 2 + y 2 2 ( z i q ) ] .
θ = Θ ¯ ( θ , ϕ ) , ϕ′ = Φ ¯ ( θ , ϕ ) .
Y 0,0 ( θ ) 2 dΩ′ = A 0,0 ( θ ) 2 .
sin θ d θ = 2 q sin h 2 q exp ( 2 q cos θ ) sin θ .
cos Θ ¯ ( θ ) = exp ( 2 q cos θ ) cosh 2 q sinh 2 q .
A l , m ( u ; q ) = q ( 2 l + 1 ) ( l m ) ! 2 π sinh 2 q ( l + m ) ! exp ( q cos θ ) P l ( m ) [ cos Θ ¯ ( θ ) ] exp ( i ) .
A l , m ( u ; q ) q ( 2 l + 1 ) ( l m ) ! 2 π ( l + m ) ! η P l ( m ) ( 1 + 2 η 2 ) exp ( i ) ,
A l , m ( u ; q ) B l , m ( q θ , ϕ ) ,
B l , m ( τ ; ϕ ) = q ( 2 l + 1 ) ( l m ) ! 2 π ( l + m ) ! exp ( τ 2 / 2 ) P l ( m ) [ 2 exp ( τ 2 ) 1 ] exp ( i ) .
ε = 1 1 N 2 l = 0 l max c l 2 ,
U l , m ( r ; q ) = 4 π A l , m ( u ; q ) exp ( i u · r ) ·
A l , 0 ( u ; q ) = q ( 2 l + 1 ) 2 π sin h 2 q exp ( q cos θ ) P l [ exp ( 2 q cos θ ) cosh 2 q sin h 2 q ] .
U l , 0 ( r ; q ) = 2 l + 1 n = 0 l l , n U 0,0 [ r ; ( 2 n + 1 ) q ] ,
l , n ( q ) = coef [ P l ( μ cosh 2 q sinh 2 q ) , μ n ] ,
c l ( q ) = 2 π 0 π A l ( θ ; q ) A ( θ ) sin θ .
cos [ Θ ( θ ) ] = 1 2 q log ( cosh 2 q + cos θ sinh 2 q ) .
c l ( q ) = ( 2 l + 1 ) π 0 π A ( θ ) P l ( θ ) sin θ d θ ,
A ( θ ) = A [ Θ ( θ ) ] 2 q ( cos θ + coth 2 q ) .
U ( r ) = l = 0 n = 0 l ( 2 l + 1 ) π c l l , n U 0,0 [ r ; ( 2 n + 1 ) q ] .
A FT ( θ ) = exp { [ w 0 2 ( 1 cos θ ) ] 8 } ,
A ( θ ) = A FT ( θ ) exp [ i s ( θ 4 2 cos θ ) ] ,

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