Abstract

The monochromatic nonparaxial vector fields that achieve a minimum spatial spread for a given directional spread are found. The derivation of these fields is analogous to the one presented in part I of this series for the case of scalar fields. This derivation is based on a variational treatment and multipolar expansion. The resulting lower bounds for the spreads of vector fields turn out to be considerably more restrictive than for scalar fields.

© 2006 Optical Society of America

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References

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  1. M. A. Alonso, R. Borghi, and M. Santarsiero, 'Nonparaxial fields with maximum joint spatial-directional localization. I. Scalar case,' J. Opt. Soc. Am. A 23, 691-700 (2006).
    [CrossRef]
  2. M. A. Alonso and G. W. Forbes, 'Uncertainty products for nonparaxial wave fields,' J. Opt. Soc. Am. A 17, 2391-2402 (2000).
    [CrossRef]
  3. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 13.2.1.
  4. J. D. Jackson, Classical Electrodynamics (Wiley, 1998), pp. 744-758.
  5. A. J. Devaney and E. Wolf, 'Multipole expansions and plane wave representations of the electromagnetic field,' J. Math. Phys. 15, 234-244 (1974).
    [CrossRef]
  6. C. J. R. Sheppard, 'Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,' IEE J. Microwaves, Opt. Acoust. 2, 163-166 (1978).
    [CrossRef]
  7. C. J. R. Sheppard and K. Larkin, 'Optimal concentration of electromagnetic radiation,' J. Mod. Opt. 41, 1495-1505 (1994).
    [CrossRef]
  8. J. D. Lawrence, A Catalog of Special Plane Curves (Dover, 1972).
  9. R. H. Jordan and D. G. Hall, 'Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,' Opt. Lett. 19, 427-429 (1994).
    [CrossRef] [PubMed]
  10. D. G. Hall, 'Vector-beam solutions of Maxwell's wave equation,' Opt. Lett. 21, 9-11 (1996).
    [CrossRef] [PubMed]
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    [CrossRef]
  12. P. L. Greene and D. G. Hall, 'Properties and diffraction of vector Bessel-Gauss beams,' J. Opt. Soc. Am. A 15, 3020-3027 (1998).
    [CrossRef]
  13. C. J. R. Sheppard and S. Saghafi, 'Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,' Opt. Lett. 24, 1543-1545 (1999).
    [CrossRef]
  14. K. S. Youngworth and T. G. Brown, 'Focusing of high numerical aperture cylindrical vector beams,' Opt. Express 7, 77-87 (2000).
    [CrossRef] [PubMed]
  15. J. Lekner, 'Invariants of three types of generalized Bessel beams,' J. Opt. 6, 837-843 (2004).
    [CrossRef]
  16. F. Gori, 'Polarization basis for vortex beams,' J. Opt. Soc. Am. A 18, 1612-1616 (2001).
    [CrossRef]
  17. R. Borghi and M. Santarsiero, 'Nonparaxial propagation of spirally polarized optical beams,' J. Opt. Soc. Am. A 21, 2029-2037 (2004).
    [CrossRef]
  18. R. Borghi, M. Santarsiero, and M. A. Alonso, 'Highly focused spirally polarized beams,' J. Opt. Soc. Am. A 22, 1420-1431 (2005).
    [CrossRef]
  19. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, 'Longitudinal field modes probed by single molecules,' Phys. Rev. Lett. 86, 5251-5254 (2001).
    [CrossRef] [PubMed]
  20. D. P. Biss and T. G. Brown, 'Polarization-vortex-driven second-harmonic generation,' Opt. Lett. 28, 923-925 (2003).
    [CrossRef] [PubMed]
  21. D. P. Biss and T. G. Brown, 'Cylindrical vector beam focusing through a dielectric interface,' Opt. Express 9, 490-497 (2001).
    [CrossRef] [PubMed]
  22. See Ref. , p. 98.

2006 (1)

2005 (1)

2004 (2)

2003 (1)

2001 (3)

2000 (2)

1999 (1)

1998 (1)

1996 (2)

1994 (2)

C. J. R. Sheppard and K. Larkin, 'Optimal concentration of electromagnetic radiation,' J. Mod. Opt. 41, 1495-1505 (1994).
[CrossRef]

R. H. Jordan and D. G. Hall, 'Free-space azimuthal paraxial wave equation: the azimuthal Bessel-Gauss beam solution,' Opt. Lett. 19, 427-429 (1994).
[CrossRef] [PubMed]

1978 (1)

C. J. R. Sheppard, 'Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,' IEE J. Microwaves, Opt. Acoust. 2, 163-166 (1978).
[CrossRef]

1974 (1)

A. J. Devaney and E. Wolf, 'Multipole expansions and plane wave representations of the electromagnetic field,' J. Math. Phys. 15, 234-244 (1974).
[CrossRef]

Alonso, M. A.

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, 'Longitudinal field modes probed by single molecules,' Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

Biss, D. P.

Borghi, R.

Born, M.

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

Brown, T. G.

Devaney, A. J.

A. J. Devaney and E. Wolf, 'Multipole expansions and plane wave representations of the electromagnetic field,' J. Math. Phys. 15, 234-244 (1974).
[CrossRef]

Forbes, G. W.

Gori, F.

Greene, P. L.

Hall, D. G.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998), pp. 744-758.

Jordan, R. H.

Larkin, K.

C. J. R. Sheppard and K. Larkin, 'Optimal concentration of electromagnetic radiation,' J. Mod. Opt. 41, 1495-1505 (1994).
[CrossRef]

Lawrence, J. D.

J. D. Lawrence, A Catalog of Special Plane Curves (Dover, 1972).

Lekner, J.

J. Lekner, 'Invariants of three types of generalized Bessel beams,' J. Opt. 6, 837-843 (2004).
[CrossRef]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, 'Longitudinal field modes probed by single molecules,' Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

Saghafi, S.

Santarsiero, M.

Sheppard, C. J. R.

C. J. R. Sheppard and S. Saghafi, 'Transverse-electric and transverse-magnetic beam modes beyond the paraxial approximation,' Opt. Lett. 24, 1543-1545 (1999).
[CrossRef]

C. J. R. Sheppard and K. Larkin, 'Optimal concentration of electromagnetic radiation,' J. Mod. Opt. 41, 1495-1505 (1994).
[CrossRef]

C. J. R. Sheppard, 'Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,' IEE J. Microwaves, Opt. Acoust. 2, 163-166 (1978).
[CrossRef]

Wolf, E.

A. J. Devaney and E. Wolf, 'Multipole expansions and plane wave representations of the electromagnetic field,' J. Math. Phys. 15, 234-244 (1974).
[CrossRef]

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

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, 'Longitudinal field modes probed by single molecules,' Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

K. S. Youngworth and T. G. Brown, 'Focusing of high numerical aperture cylindrical vector beams,' Opt. Express 7, 77-87 (2000).
[CrossRef] [PubMed]

IEE J. Microwaves, Opt. Acoust. (1)

C. J. R. Sheppard, 'Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,' IEE J. Microwaves, Opt. Acoust. 2, 163-166 (1978).
[CrossRef]

J. Math. Phys. (1)

A. J. Devaney and E. Wolf, 'Multipole expansions and plane wave representations of the electromagnetic field,' J. Math. Phys. 15, 234-244 (1974).
[CrossRef]

J. Mod. Opt. (1)

C. J. R. Sheppard and K. Larkin, 'Optimal concentration of electromagnetic radiation,' J. Mod. Opt. 41, 1495-1505 (1994).
[CrossRef]

J. Opt. (1)

J. Lekner, 'Invariants of three types of generalized Bessel beams,' J. Opt. 6, 837-843 (2004).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Express (2)

Opt. Lett. (4)

Phys. Rev. Lett. (1)

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, 'Longitudinal field modes probed by single molecules,' Phys. Rev. Lett. 86, 5251-5254 (2001).
[CrossRef] [PubMed]

Other (4)

J. D. Lawrence, A Catalog of Special Plane Curves (Dover, 1972).

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

J. D. Jackson, Classical Electrodynamics (Wiley, 1998), pp. 744-758.

See Ref. , p. 98.

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Figures (6)

Fig. 1
Fig. 1

Behavior of the minimum eigenvalue Λ min as a function of w for several values of η m . Solid curves refer to η m > 0 , while dashed curves refer to η m < 0 . The case m = 0 is plotted as a thick solid curve.

Fig. 2
Fig. 2

(a) Flux lines of the unit vectors e 1 (solid curves) and e 2 (dotted curves) over the sphere of directions. (b) Stereographic projection of the flux lines of the unit vectors e 1 (solid curves) and e 2 (dotted curves) near the south pole ( θ π ) .

Fig. 3
Fig. 3

Behavior of the magnitude of the angular spectra of the MUFs for several values of w.

Fig. 4
Fig. 4

Lower bound for the spreads of vector fields, composed of a segment corresponding to the spreads of the MUFs with η m = 1 (thick solid curve) and a vertical segment corresponding to a linear combination of the MUF with η m = 1 , w = and the field with m = 0 , w = (thick dashed curve). The thin dashed curve corresponds to the lower bound for fields with m = 0 . The straight diagonal dotted line corresponds to the algebraic lower bound in Eq. (19).

Fig. 5
Fig. 5

Plots of the modulus (left column) and field lines (middle column) of the transverse component of the focal electric field produced by vectorial MUFs across the plane z = 0 , for w = 1 5 (first row), w = 1 2 (second row), and w = 10 (third row) all for γ = 1 and δ = 0 . The modulus of the longitudinal component of the electric field for these cases is given in the right column.

Fig. 6
Fig. 6

Plots of the total electric intensity across the x , z plane (left column) and the y , z plane (right column) for w = 1 2 (first row), w = 1 5 (second row), and w = 10 (third row).

Equations (110)

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E ( r ) = 4 π A ( u ) exp ( i k u r ) d Ω ,
c B ( r ) = 4 π u × A ( u ) exp ( i u r ) d Ω ,
E 1 , E 2 = 4 π A 1 * ( u ) A 2 ( u ) d Ω ,
F 1 , F 2 = 1 2 [ A 1 * A 2 + ( A 1 × u ) * ( A 2 × u ) ] d Ω ,
O F = F , O F F 2 ,
F , O F = 1 2 4 π [ A * O A + ( A × u ) * O ( A × u ) ] d Ω .
O E = E , O E E 2 ,
E , O E = 4 π A * O A d Ω ,
( A × u ) * O ( A × u ) = A * O A A * u × ( [ O , u ] × A ) ,
F , O F = E , O E 1 2 4 π A * u × ( [ O , u ] × A ) d Ω .
Δ θ = arccos u E ,
u E = 4 π u A ( u ) 2 d Ω E 2 = 4 π cos θ A ( u ) 2 d Ω E 2 e z ,
u F = u E .
L = i u × u ,
L = i e θ sin θ ϕ i e ϕ θ ,
Δ r = L 2 E 1 2 .
L 2 F = L 2 E .
Δ r tan Δ θ 1 .
Δ θ + δ r π 2 ,
[ δ δ A * ( ) ] j = δ δ A j * ( ) .
δ δ A * 4 π A * ( u ) O A ( u ) d Ω = P u O A ( u ) ,
δ O E δ A * = δ δ A * E , O E E 2 = P u O O E E 2 A ( u ) .
C 1 δ Δ r δ A * + C 2 δ Δ θ δ A * = 0 ,
2 w 2 ( 1 cos θ ) A ( u ) + w 2 P u L 2 A ( u ) = Λ A ( u ) ,
w 4 = C 2 sin Δ θ C 1 Δ r ,
Λ = 2 ( 1 cos Δ θ ) w 2 + Δ r 2 w 2 .
A ( u ) = l = 1 m = l + l [ a l , m Y l , m ( u ) + b l , m Z l , m ( u ) ] ,
Y l , m ( u ) = L Y l , m ( u ) .
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 ) d Ω = δ l , l δ m , m l ( l + 1 ) ,
4 π Z l , m * ( u ) Z l , m ( u ) d Ω = δ l , l δ m , m l ( l + 1 ) ,
4 π Y l , m * ( u ) Z l , m ( u ) d Ω = 0 ,
L 2 Z l , m = l ( l + 1 ) Z l , m 2 i l ( l + 1 ) u Y l , m ,
4 π Y l , m * cos θ Y l , m d Ω = 4 π Z l , m * cos θ Z l , m d Ω = δ m , m ( X l + 1 , m δ l , l + 1 + X l , m δ l , l 1 ) ,
4 π Y l , m * cos θ Z l , m d Ω = i m δ m , m δ l , l ,
X l , m = ( l 2 1 ) l 2 m 2 4 l 2 1 .
[ 1 + w 4 2 l ( l + 1 ) ] a l , m X l + 1 , m l ( l + 1 ) a l + 1 , m X l , m l ( l + 1 ) a l 1 , m + i m b l , m l ( l + 1 ) = Λ ¯ a l , m ,
[ 1 + w 4 2 l ( l + 1 ) ] b l , m X l + 1 , m l ( l + 1 ) b l + 1 , m X l , m l ( l + 1 ) b l 1 , m i m a l , m l ( l + 1 ) = Λ ¯ b l , m ,
U l , m ± ( u ) = Y l , m ( u ) ± i Z l , m ( u ) 2 .
4 π [ U l , m η ( u ) ] * U l , m η ( u ) d Ω = δ η , η δ l , l δ m , m l ( l + 1 ) ,
4 π [ U l , m η ( u ) ] * cos θ U l , m η ( u ) d Ω = δ η , η δ m , m
× ( X l + 1 , m δ l l + 1 + X l , m δ l , l 1 + η m δ l , l ) .
P u L 2 U l , m ± = l ( l + 1 ) U l , m ± .
A ( u ) = l = 1 m = l l η = ± c l , m η U l , m η ( u ) ,
[ 1 + w 4 2 l ( l + 1 ) m l ( l + 1 ) ] c l , m ± X l + 1 , m l ( l + 1 ) c l + 1 , m ± X l , m l ( l + 1 ) c l 1 , m ± = Λ ¯ c l , m ± .
A ( u ) = l = 1 c l [ α U l , 1 + ( u ) + β U l , 1 ( u ) ] ,
A ( u ) = A ( θ ) ( γ e 1 + δ e 2 ) ,
α = γ i δ 2 ,
β = γ + i δ 2 .
e 1 = cos ϕ e θ sin ϕ e ϕ ,
e 2 = sin ϕ e θ + cos ϕ e ϕ ,
A ( θ ) = l = 1 c l F l ( cos θ ) ,
F l ( t ) = 2 l + 1 4 π 1 l ( l + 1 ) [ ( 1 t ) P l ( t ) + l ( l + 1 ) P l ( t ) ] ,
Δ r 2 = l = 1 l 2 ( l + 1 ) 2 c l 2 l = 1 l ( l + 1 ) c l 2 ,
cos Δ θ = l = 1 [ 2 Re { c l * c l + 1 X l + 1 , 1 } + c l 2 ] l = 1 l ( l + 1 ) c l 2 ,
A ( u ) = l = 1 c l [ α U l , 0 + ( u ) + β U l , 0 ( u ) ] ,
U l , 0 ± ( u ) = 2 l + 1 4 π P l ( 1 ) ( cos θ ) ( e θ ± i e ϕ ) .
A ( u ) = A ( θ ) ( γ e θ + δ e ϕ ) ,
A ( θ ) = l = 1 c l 2 l + 1 4 π P l ( 1 ) ( cos θ ) .
P u L 2 A ( u ) = Δ r 2 A ( u ) .
A ( u ) A ( θ ) ( γ e x + δ e y ) ,
A ( θ ) A 0 exp ( θ 2 2 w 2 ) ,
Δ θ w ,
δ r π 2 w π 2 Δ θ ,
A ( θ ) A 0 θ exp ( θ 2 2 w 2 ) ,
N l , m ( r ) = i × [ r Π l , m ( r ) ] ,
M l , m ( r ) = 1 i × N l , , m ( r ) = × { × [ r Π l , m ( r ) ] } ,
Π l , m ( r ) = j l ( r ) Y l , m ( θ , ϕ ) ,
N l , m ( r ) = ( i ) l 4 π 4 π Y l , m ( u ) exp ( i u r ) d Ω ,
M l , m ( r ) = ( i ) l 4 π 4 π Z l , m ( u ) exp ( i u r ) d Ω ,
N l , m ( r ) = j l ( r ) Y l , m ( θ , ϕ ) ,
M l , m ( r ) = e r j l ( r ) r l ( l + 1 ) Y l , m ( θ , ϕ ) + i [ l j l ( r ) r j l 1 ( r ) ] Z l , m ( θ , ϕ ) .
[ L i , u j ] = i ϵ i j n u n ,
[ L 2 , L ] = 0 ,
[ L i , L j ] = i ϵ i j n L n ,
L i u j = i ϵ i j n u n + u j L i .
{ [ L 2 , u ] } j = L i ( L i u j ) u j L i L j = L i ( i ϵ i j n u n + u j L i ) u j L i L i = i ϵ i j n ( L i u n ) + ( L i u j ) L i u j L i L i = i ϵ i j n ( i ϵ i n m u m + u n L i ) + ( i ϵ i j n u n + u j L i ) L i u j L i L i = 2 u j + 2 i ϵ i j n u n L i = { 2 u + 2 i u × L } j ,
u × ( [ L 2 , u ] × A ) = 2 u × [ u × A + i ( u × L ) × A ] .
( u × L ) × A = i u × A u ( L A ) ,
u × ( [ L 2 , u ] × A ) = 0 .
L 2 Z = L 2 ( u × L ) Y = ( 2 u + 2 i u × L + u L 2 ) × L Y = 2 u × L Y + 2 i ( u × L ) × L Y + u × L L 2 Y .
{ ( u × L ) × L Y } n = ϵ i m n ( ϵ i j k u j L k ) L m Y = u j ( L n L j ) Y u n L j L j Y = u j ( i ϵ n j i L i + L j L n ) u n L j L j Y = { i u × L Y u L 2 Y } n ,
L 2 Z = 2 i u L 2 Y + u × L L 2 Y = l ( l + 1 ) ( Z 2 i u Y ) ,
4 π cos θ Y l , m * Y l , m d Ω ,
4 π cos θ Z l , m * Z l , m d Ω ,
4 π cos θ Y l , m * Z l , m d Ω .
4 π ( L Y l , m ) * ( u z L Y l , m ) d Ω ,
4 π Y l , m * L u z L Y l , m d Ω .
L u z L = i e z u × L + u z L 2 .
4 π Y l , m * L u z L Y l , m d Ω = l ( l + 1 ) 4 π Y l , m * cos θ Y l , m d Ω + 4 π Y l , m * i e z u × L Y l , m d Ω .
4 π Y l , m * cos θ Y l , m d Ω = ( l + 1 ) 2 m 2 ( 2 l + 1 ) ( 2 l + 3 ) δ l , l + 1 + l 2 m 2 ( 2 l + 1 ) ( 2 l 1 ) δ l , l 1 .
i e z u × L = sin θ θ ,
4 π Y l , m * i e z u × L Y l , m d Ω = 4 π Y l , m * sin θ Y l , m θ d Ω = ( 2 l + 1 ) ( 2 l + 1 ) 2 ( l m ) ! ( l m ) ! ( l + m ) ! ( l + m ) ! δ m , m 1 + 1 ( 1 t 2 ) P l ( m ) d P l ( m ) d t d t ,
( 1 t 2 ) d P l ( m ) d t = l ( l m + 1 ) 2 l + 1 P l + 1 ( m ) + ( l + 1 ) ( l + m ) 2 l + 1 P l 1 ( m ) ,
1 + 1 P l ( m ) P l ( m ) d t = 2 2 l + 1 ( l + m ) ! ( l m ) ! δ l , l ,
4 π Y l , m * i e z u × L Y l , m d Ω = ( l + 2 ) ( l + 1 ) 2 m 2 ( 2 l + 1 ) ( 2 l + 3 ) δ l , l + 1 + ( l 1 ) l 2 m 2 ( 2 l + 1 ) ( 2 l 1 ) δ l , l 1 .
4 π cos θ Y l , m * Y l , m d Ω = l ( l + 2 ) ( l + 1 ) 2 m 2 ( 2 l + 1 ) ( 2 l + 3 ) δ l , l + 1 + ( l 1 ) ( l + 1 ) l 2 m 2 ( 2 l + 1 ) ( 2 l 1 ) δ l , l 1 .
4 π cos θ Y l , m * Z l , m d Ω = 4 π Y l , m * L ( u z u × L ) Y l , m d Ω .
L ( u z u × L ) = i e z u × ( u × L ) + u z L ( u × L ) = i L z + i u L = i L z ,
4 π cos θ Y l , m * Z l , m d Ω = i 4 π Y l , m * L z Y l , m d Ω = i m δ m , m δ l , l .
Y l , ± 1 = exp ( ± i ϕ ) 2 l + 1 4 π 1 l ( l + 1 ) × { e θ P l ( cos θ ) i e ϕ [ cos θ P l ( cos θ ) l ( l + 1 ) P l ( cos θ ) ] } ,
P l ( 1 ) ( x ) = 1 x 2 d P l ( x ) d x ,
1 x 2 d P l ( 1 ) d x x d P l d x + ( 1 x 2 ) d 2 P l d x 2 = x d P l d x l ( l + 1 ) P l ( x ) .
Z l , ± 1 = exp ( ± i ϕ ) 2 l + 1 4 π 1 l ( l + 1 ) { e ϕ P l ( cos θ ) ± i e θ [ cos θ P l ( cos θ ) l ( l + 1 ) P l ( cos θ ) ] } .
U l , ± 1 ± = F l ( cos θ ) exp ( ± i ϕ ) ( e θ ± i e ϕ ) ,
N l , m ( r ) = j l ( r ) Y l , m ( θ , ϕ ) ,
M l , m ( r ) = 1 i j l ( r ) Z l , m ( θ , ϕ ) + 1 i j l ( r ) × Y l , m ( θ , ϕ ) .
i × L = r 2 ( 1 + r r ) ,
M l , m ( r ) = e r j l ( r ) r l ( l + 1 ) Y l , m ( θ , ϕ ) + [ j l ( r ) Y l , m ( θ , ϕ ) i j l ( r ) Z l , m ( θ , ϕ ) ] .
Y l , m ( θ , ϕ ) = 1 r u Y l , m ( θ , ϕ ) = i r Z l , m ( θ , ϕ ) ,

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