Abstract

In paraxial optics, the spatial and angular localization of a beam are usually characterized through second moments in intensity. For these measures, Gaussian beams have the property of achieving a minimum angular spread for a given spatial spread (or beam waist). For wide-angle fields, however, the standard measures of spatial and angular localization become inappropriate, and new definitions must be used. Previously proposed definitions [J. Opt. Soc. Am. A 17, 2391 (2000) ] are adopted, and the scalar monochromatic wave fields that achieve a minimum angular spread for a given spatial spread are found.

© 2006 Optical Society of America

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References

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  1. A. E. Siegman, 'New developments in laser resonators,' in Optical Resonators, D.A.Holmes, ed., Proc. SPIE 1224, 2-14 (1990).
  2. M. A. Porras, 'The best quality optical beam beyond the paraxial approximation,' Opt. Commun. 111, 338-349 (1994).
    [CrossRef]
  3. M. A. Alonso and G. W. Forbes, 'Uncertainty products for nonparaxial wave fields,' J. Opt. Soc. Am. A 17, 2391-2402 (2000).
    [CrossRef]
  4. T. Opatrný, 'Mean value and uncertainty in optical phase--a simple mechanical analogy,' J. Phys. A 27, 7201-7208 (1994).
    [CrossRef]
  5. G. W. Forbes and M. A. Alonso, 'Measures of spread for periodic distributions and the associated uncertainty relations,' Am. J. Phys. 69, 340-347 (2001).
    [CrossRef]
  6. R. Jackiw, 'Minimum uncertainty product, number-phase uncertainty product, and coherent states,' J. Math. Phys. 9, 339-346 (1968).
    [CrossRef]
  7. P. Carruthers and M. M. Nieto, 'Phase and angle variables in quantum mechanics,' Rev. Mod. Phys. 40, 411-440 (1968).
    [CrossRef]
  8. A. Luks and V. Perinová, 'Extended number state basis and number-phase intelligent states of light fields. I. Mapping and operator ordering approach to quantum phase problems,' Czech. J. Phys. 41, 1205-1230 (1991).
    [CrossRef]
  9. T. Opatrný, 'Number-phase uncertainty relations,' J. Phys. A 28, 6961-6975 (1995).
    [CrossRef]
  10. G. W. Forbes, M. A. Alonso, and A. E. Siegman, 'Uncertainty relations and minimum-uncertainty states for the discrete Fourier transform and the Fourier series,' J. Phys. A 36, 7027-7047 (2003).
    [CrossRef]
  11. S. Steinberg and K. B. Wolf, 'Invariant inner products on spaces of solutions of the Klein-Gordon and Helmholtz equations,' J. Math. Phys. 22, 1660-1663 (1981).
    [CrossRef]
  12. M. A. Alonso, 'Measurement of Helmholtz wave fields,' J. Opt. Soc. Am. A 17, 1256-1264 (2000).
    [CrossRef]
  13. J. Durnin, 'Exact solutions for nondiffracting beams,' J. Opt. Soc. Am. A 4, 651-654 (1987).
    [CrossRef]
  14. J. C. Gutiérrez-Vega and M. A. Bandres, 'Helmholtz-Gauss waves,' J. Opt. Soc. Am. A 22, 289-298 (2005).
    [CrossRef]
  15. N. W. McLachlan, Theory and Applications of Mathieu Functions (Dover, 1964), p. 10.
  16. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Harcourt, 2001), p. 787.
  17. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).
  18. R. Borghi, 'On the angular-spectrum representation of multipole wavefields,' J. Opt. Soc. Am. A 21, 1805-1810 (2004).
    [CrossRef]
  19. C. J. R. Sheppard, 'High-aperture beams,' J. Opt. Soc. Am. A 18, 1579-1587 (2001).
    [CrossRef]
  20. M. Alonso, R. Borghi, and M. Santarsiero, 'Joint spatial-directional localization features of wave fields focused at a complex point,' J. Opt. Soc. Am. A (to be published).
  21. M. Alonso, R. Borghi, and M. Santarsiero, 'Nonparaxial fields with maximum joint spatial-directional localization. II. Vectorial case,' J. Opt. Soc. Am. A 23, 701-712 (2006).
    [CrossRef]

2006 (1)

2005 (1)

2004 (1)

2003 (1)

G. W. Forbes, M. A. Alonso, and A. E. Siegman, 'Uncertainty relations and minimum-uncertainty states for the discrete Fourier transform and the Fourier series,' J. Phys. A 36, 7027-7047 (2003).
[CrossRef]

2001 (2)

G. W. Forbes and M. A. Alonso, 'Measures of spread for periodic distributions and the associated uncertainty relations,' Am. J. Phys. 69, 340-347 (2001).
[CrossRef]

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

2000 (2)

1995 (1)

T. Opatrný, 'Number-phase uncertainty relations,' J. Phys. A 28, 6961-6975 (1995).
[CrossRef]

1994 (2)

M. A. Porras, 'The best quality optical beam beyond the paraxial approximation,' Opt. Commun. 111, 338-349 (1994).
[CrossRef]

T. Opatrný, 'Mean value and uncertainty in optical phase--a simple mechanical analogy,' J. Phys. A 27, 7201-7208 (1994).
[CrossRef]

1991 (1)

A. Luks and V. Perinová, 'Extended number state basis and number-phase intelligent states of light fields. I. Mapping and operator ordering approach to quantum phase problems,' Czech. J. Phys. 41, 1205-1230 (1991).
[CrossRef]

1987 (1)

1981 (1)

S. Steinberg and K. B. Wolf, 'Invariant inner products on spaces of solutions of the Klein-Gordon and Helmholtz equations,' J. Math. Phys. 22, 1660-1663 (1981).
[CrossRef]

1968 (2)

R. Jackiw, 'Minimum uncertainty product, number-phase uncertainty product, and coherent states,' J. Math. Phys. 9, 339-346 (1968).
[CrossRef]

P. Carruthers and M. M. Nieto, 'Phase and angle variables in quantum mechanics,' Rev. Mod. Phys. 40, 411-440 (1968).
[CrossRef]

Alonso, M.

M. Alonso, R. Borghi, and M. Santarsiero, 'Nonparaxial fields with maximum joint spatial-directional localization. II. Vectorial case,' J. Opt. Soc. Am. A 23, 701-712 (2006).
[CrossRef]

M. Alonso, R. Borghi, and M. Santarsiero, 'Joint spatial-directional localization features of wave fields focused at a complex point,' J. Opt. Soc. Am. A (to be published).

Alonso, M. A.

G. W. Forbes, M. A. Alonso, and A. E. Siegman, 'Uncertainty relations and minimum-uncertainty states for the discrete Fourier transform and the Fourier series,' J. Phys. A 36, 7027-7047 (2003).
[CrossRef]

G. W. Forbes and M. A. Alonso, 'Measures of spread for periodic distributions and the associated uncertainty relations,' Am. J. Phys. 69, 340-347 (2001).
[CrossRef]

M. A. Alonso and G. W. Forbes, 'Uncertainty products for nonparaxial wave fields,' J. Opt. Soc. Am. A 17, 2391-2402 (2000).
[CrossRef]

M. A. Alonso, 'Measurement of Helmholtz wave fields,' J. Opt. Soc. Am. A 17, 1256-1264 (2000).
[CrossRef]

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Harcourt, 2001), p. 787.

Bandres, M. A.

Borghi, R.

Carruthers, P.

P. Carruthers and M. M. Nieto, 'Phase and angle variables in quantum mechanics,' Rev. Mod. Phys. 40, 411-440 (1968).
[CrossRef]

Durnin, J.

Forbes, G. W.

G. W. Forbes, M. A. Alonso, and A. E. Siegman, 'Uncertainty relations and minimum-uncertainty states for the discrete Fourier transform and the Fourier series,' J. Phys. A 36, 7027-7047 (2003).
[CrossRef]

G. W. Forbes and M. A. Alonso, 'Measures of spread for periodic distributions and the associated uncertainty relations,' Am. J. Phys. 69, 340-347 (2001).
[CrossRef]

M. A. Alonso and G. W. Forbes, 'Uncertainty products for nonparaxial wave fields,' J. Opt. Soc. Am. A 17, 2391-2402 (2000).
[CrossRef]

Gutiérrez-Vega, J. C.

Jackiw, R.

R. Jackiw, 'Minimum uncertainty product, number-phase uncertainty product, and coherent states,' J. Math. Phys. 9, 339-346 (1968).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

Luks, A.

A. Luks and V. Perinová, 'Extended number state basis and number-phase intelligent states of light fields. I. Mapping and operator ordering approach to quantum phase problems,' Czech. J. Phys. 41, 1205-1230 (1991).
[CrossRef]

McLachlan, N. W.

N. W. McLachlan, Theory and Applications of Mathieu Functions (Dover, 1964), p. 10.

Nieto, M. M.

P. Carruthers and M. M. Nieto, 'Phase and angle variables in quantum mechanics,' Rev. Mod. Phys. 40, 411-440 (1968).
[CrossRef]

Opatrný, T.

T. Opatrný, 'Number-phase uncertainty relations,' J. Phys. A 28, 6961-6975 (1995).
[CrossRef]

T. Opatrný, 'Mean value and uncertainty in optical phase--a simple mechanical analogy,' J. Phys. A 27, 7201-7208 (1994).
[CrossRef]

Perinová, V.

A. Luks and V. Perinová, 'Extended number state basis and number-phase intelligent states of light fields. I. Mapping and operator ordering approach to quantum phase problems,' Czech. J. Phys. 41, 1205-1230 (1991).
[CrossRef]

Porras, M. A.

M. A. Porras, 'The best quality optical beam beyond the paraxial approximation,' Opt. Commun. 111, 338-349 (1994).
[CrossRef]

Santarsiero, M.

M. Alonso, R. Borghi, and M. Santarsiero, 'Nonparaxial fields with maximum joint spatial-directional localization. II. Vectorial case,' J. Opt. Soc. Am. A 23, 701-712 (2006).
[CrossRef]

M. Alonso, R. Borghi, and M. Santarsiero, 'Joint spatial-directional localization features of wave fields focused at a complex point,' J. Opt. Soc. Am. A (to be published).

Sheppard, C. J. R.

Siegman, A. E.

G. W. Forbes, M. A. Alonso, and A. E. Siegman, 'Uncertainty relations and minimum-uncertainty states for the discrete Fourier transform and the Fourier series,' J. Phys. A 36, 7027-7047 (2003).
[CrossRef]

A. E. Siegman, 'New developments in laser resonators,' in Optical Resonators, D.A.Holmes, ed., Proc. SPIE 1224, 2-14 (1990).

Steinberg, S.

S. Steinberg and K. B. Wolf, 'Invariant inner products on spaces of solutions of the Klein-Gordon and Helmholtz equations,' J. Math. Phys. 22, 1660-1663 (1981).
[CrossRef]

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Harcourt, 2001), p. 787.

Wolf, K. B.

S. Steinberg and K. B. Wolf, 'Invariant inner products on spaces of solutions of the Klein-Gordon and Helmholtz equations,' J. Math. Phys. 22, 1660-1663 (1981).
[CrossRef]

Am. J. Phys. (1)

G. W. Forbes and M. A. Alonso, 'Measures of spread for periodic distributions and the associated uncertainty relations,' Am. J. Phys. 69, 340-347 (2001).
[CrossRef]

Czech. J. Phys. (1)

A. Luks and V. Perinová, 'Extended number state basis and number-phase intelligent states of light fields. I. Mapping and operator ordering approach to quantum phase problems,' Czech. J. Phys. 41, 1205-1230 (1991).
[CrossRef]

J. Math. Phys. (2)

R. Jackiw, 'Minimum uncertainty product, number-phase uncertainty product, and coherent states,' J. Math. Phys. 9, 339-346 (1968).
[CrossRef]

S. Steinberg and K. B. Wolf, 'Invariant inner products on spaces of solutions of the Klein-Gordon and Helmholtz equations,' J. Math. Phys. 22, 1660-1663 (1981).
[CrossRef]

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

J. Phys. A (3)

T. Opatrný, 'Mean value and uncertainty in optical phase--a simple mechanical analogy,' J. Phys. A 27, 7201-7208 (1994).
[CrossRef]

T. Opatrný, 'Number-phase uncertainty relations,' J. Phys. A 28, 6961-6975 (1995).
[CrossRef]

G. W. Forbes, M. A. Alonso, and A. E. Siegman, 'Uncertainty relations and minimum-uncertainty states for the discrete Fourier transform and the Fourier series,' J. Phys. A 36, 7027-7047 (2003).
[CrossRef]

Opt. Commun. (1)

M. A. Porras, 'The best quality optical beam beyond the paraxial approximation,' Opt. Commun. 111, 338-349 (1994).
[CrossRef]

Rev. Mod. Phys. (1)

P. Carruthers and M. M. Nieto, 'Phase and angle variables in quantum mechanics,' Rev. Mod. Phys. 40, 411-440 (1968).
[CrossRef]

Other (5)

N. W. McLachlan, Theory and Applications of Mathieu Functions (Dover, 1964), p. 10.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 5th ed. (Harcourt, 2001), p. 787.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

A. E. Siegman, 'New developments in laser resonators,' in Optical Resonators, D.A.Holmes, ed., Proc. SPIE 1224, 2-14 (1990).

M. Alonso, R. Borghi, and M. Santarsiero, 'Joint spatial-directional localization features of wave fields focused at a complex point,' J. Opt. Soc. Am. A (to be published).

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Figures (8)

Fig. 1
Fig. 1

Geometric interpretation of the angular spread measure in terms of u U , the centroid of the distribution A ( u ) 2 over the unit sphere of directions. Δ θ corresponds to the half-angle of the cone subtended from the origin by the intersection of the unit sphere of directions and a plane containing this centroid and perpendicular to the line joining the origin and the centroid.

Fig. 2
Fig. 2

Behavior of the minimum eigenvalue Λ min , as a function of w, for m = 0 (thick solid curve), m = 1 (thin solid curve), m = 2 (dashed curve), and m = 3 (dotted curve).

Fig. 3
Fig. 3

Behavior of the lower bound for the uncertainties pertinent to 2D and 3D scalar MUFs (solid curves), together with those corresponding to the equality sign in inequalities (23, 24) (dotted curves).

Fig. 4
Fig. 4

Behavior, as functions of w, of Δ θ (solid curve) and Δ r (dotted curve).

Fig. 5
Fig. 5

Behavior of the HWHM of scalar MUFs as a function of the parameter Δ r .

Fig. 6
Fig. 6

Polar plots of the angular spectra for several values of w.

Fig. 7
Fig. 7

Behavior of the angular spectra for several values of w.

Fig. 8
Fig. 8

Two-dimensional intensity maps evaluated across the ( z , r ) plane for MUFs having w = ( a ) 100 , (b) 0.8, (c) 0.5, (d) 0.2.

Equations (74)

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U ( r ) = 4 π A ( u ) exp ( i k u r ) d Ω ,
i r U ( r ) = 4 π u A ( u ) exp ( i u r ) d Ω .
L = i r × r .
L = i u × u ,
L = i e θ sin θ ϕ i e ϕ θ ,
U 1 , U 2 = 4 π A 1 * ( u ) A 2 ( u ) d Ω .
U 1 , U 2 = lim R 1 8 π 2 R S R U 1 * ( r ) U 2 ( r ) d 3 r ,
U = U , U 1 2 .
O U = U , O U U 2 = 1 U 2 4 π A * ( u ) O A ( u ) d Ω .
U , O U = O U , U .
δ δ A R ( u ) 4 π A R ( u ) B ( u ) d Ω = B ( u )
x j j x j b j = b j .
δ δ A R ( u ) U , O U = O A ( u ) + [ O A ( u ) ] * ,
δ δ A I ( u ) U , O U = i O A ( u ) i [ O A ( u ) ] * .
δ δ A ( u ) U , O U = [ O A ( u ) ] * ,
δ δ A * ( u ) U , O U = O A ( u ) .
δ O U δ A * = O A U 2 O U U 2 δ U 2 δ A * = O O U U 2 A ( u ) .
Δ u = 1 u U 2 .
Δ θ = arccos u U = arcsin Δ u .
Δ r = L 2 U ,
( Δ r 2 + 1 ) Δ u 2 1 .
( Δ r 2 + 1 4 ) Δ u 2 1 4 .
Δ r tan Δ θ 1
Δ r tan Δ θ 1 2
Δ θ + δ r π 2 ,
δ δ A * ( C 1 Δ θ + C 2 Δ r ) = 0 ,
δ δ A * [ Δ r + μ ( Δ θ Δ ¯ θ ) ] = 0 ,
δ Δ θ δ A * = u U u U 1 u U 2 δ u U δ A * = u U u U u u U U 2 sin Δ θ A ( u ) .
δ Δ θ δ A * = cos θ u U U 2 sin Δ θ A ( u ) = cos θ cos Δ θ U 2 sin Δ θ A ( u ) .
δ Δ r δ A * = 1 2 Δ r δ Δ r 2 δ A * = L 2 Δ r 2 2 U 2 Δ r A ( u ) .
C 1 cos θ cos Δ θ sin Δ θ A ( u ) + C 2 L 2 Δ r 2 2 Δ r A ( u ) = 0 .
w 4 = C 2 C 1 sin Δ θ Δ r ,
Λ = 2 w 2 ( 1 cos Δ θ ) + w 2 Δ r 2 ,
[ 2 w 2 ( 1 cos θ ) + w 2 L 2 ] A ( u ) = Λ A ( u ) .
A ( u ) = A 0 M C [ A ( 0 , 4 w 4 ) , 4 w 4 , θ 2 ] ,
A ( u ) = l = 0 m = l + l a l , m 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 * Y l , m d Ω = δ m , m δ l , l
4 π Y l , m * cos θ Y l , m d Ω = δ m , m ( χ l + 1 , m δ l l + 1 + χ l , m δ l , l 1 ) ,
χ l , m = l 2 m 2 ( 2 l + 1 ) ( 2 l 1 ) ,
[ 1 + w 4 2 l ( l + 1 ) ] a l , m χ l + 1 , m a l + 1 , m χ l , m a l 1 , m = Λ ¯ a l , m ,
A ( u ) A 0 ( 1 + cos θ w 4 ) ,
Λ 2 w 2 .
Δ θ arccos ( 2 w 4 1 + 3 w 8 ) arccos ( 2 3 w 4 ) ,
Δ r 2 1 + 3 w 8 2 3 w 4 .
cos Δ θ π 2 Δ θ 2 3 Δ r 2 3 δ r .
A ( u ) = A 0 exp [ Φ ( θ ) w 2 ] ,
2 2 cos θ Φ 2 w 2 ( cot θ Φ + Φ + Λ ) = 0 .
A ( u ) A 0 exp ( 4 w 2 cos θ 2 ) A 0 exp ( 4 w 2 ) exp ( θ 2 2 w 2 ) ,
Δ θ w ,
δ r π 2 w π 2 Δ θ ,
s 0 2 w 2 Δ r .
Π l , m ( r ) = j l ( r ) Y l , m ( s ) ,
4 π i l Π l , m ( r ) = 4 π Y l , m ( u ) exp ( i u r ) d Ω ,
U ( r ) = 4 π l = 0 i l a l , 0 Π l , 0 ( r ) .
S R U 1 * ( r ) U 2 ( r ) d 3 r = 4 π 4 π A 1 * ( u 1 ) A 2 ( u 2 ) I ( u 1 , u 2 , R ) d Ω 1 d Ω 2 ,
I ( u 1 , u 2 , R ) = S R exp [ i ( u 2 u 1 ) r ] d 3 r = 4 π R u 2 u 1 2 [ sin ( R u 2 u 1 ) R u 2 u 1 cos ( R u 2 u 1 ) ] .
δ ( u 1 , u 2 ) = δ ( 1 u 1 u 2 ) π ,
4 π A ( u 1 ) δ ( u 1 , u 2 ) d Ω 1 = A ( u 2 ) .
4 π I ( u 1 , u 2 , R ) d Ω 1 = S R exp ( i u 2 r ) 4 π exp ( i u 1 r ) d Ω 1 d 3 r = 4 π S R exp ( i u 2 r ) sin ( r ) r d 3 r = ( 4 π ) 2 0 R r 2 [ sin ( r ) r ] 2 d r = 8 π 2 R [ 1 sin ( 2 R ) 2 R ] .
Δ r 2 = 1 U 2 lim R 1 8 π 2 R S R ( L U ) * ( L U ) d 3 r = 1 U 2 lim R 1 8 π 2 R S R ( r × U * U ) * ( r × U * U ) I d 3 r ,
U * ( r ) U ( r ) = I ( r ) 2 + i J ( r ) ,
J ( r ) = Im [ U * ( r ) U ( r ) ] ,
Δ r 2 = 1 U 2 lim R 1 8 π 2 R ( 1 4 S R I r × g 2 d 3 r + S R I r × j 2 d 3 r ) ,
Δ B 2 Δ C 2 1 4 [ B j , C j ] U 2 ,
Δ B 2 = B B U B U B U ,
Δ C 2 = C C U C U C U .
[ L i , u j ] = i ϵ i j m u m ,
Δ u 2 Δ L 2 1 4 T i j ϵ i j m u m U 2 .
Δ u 2 Δ L 2 1 4 T i j ϵ i j z 2 u U 2 .
T = [ 0 i 0 i 0 0 0 0 1 ] ,
Δ u 2 Δ L 2 u U 2 ,
Δ u 2 ( 1 + Δ L 2 ) 1 .
( Δ r 2 + 1 ) Δ u 2 1 .

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