Abstract

A systematic study of the joint spatial–directional localization features of monochromatic wave fields focused at a complex point is presented, on the basis of recently introduced measures of spatial and directional spread for wide-angle wave fields. Such features are compared with those of a class of fields defined to achieve the theoretical minimum product of these spread measures. It is found that the two classes of fields are remarkably similar.

© 2006 Optical Society of America

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References

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  1. M. Alonso, R. Borghi, and M. Santarsiero, "Nonparaxial fields with maximum joint spatial-directional localization. I. Scalar case," J. Opt. Soc. Am. A 23, 000-000 (2006).
  2. M. Alonso, R. Borghi, and M. Santarsiero, "Nonparaxial fields with maximum joint spatial-directional localization. II. Vector case," J. Opt. Soc. Am. A 23, 000-000 (2006).
    [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. Yu. A. Kravtsov, "Complex ray and complex caustics," Radiophys. Quantum Electron. 10, 719-730 (1967).
    [CrossRef]
  5. J. Arnaud, "Degenerate optical cavities. II: Effects of misalignments," Appl. Opt. 8, 1909-1917 (1969).
    [CrossRef] [PubMed]
  6. J. B. Keller and W. Streifer, "Complex rays with an application to Gaussian beams," J. Opt. Soc. Am. 61, 40-43 (1971).
    [CrossRef]
  7. G. A. Deschamps, "Gaussian beams as a bundle of complex rays," Electron. Lett. 7, 684-685 (1971).
    [CrossRef]
  8. L. B. Felsen, "Evanescent waves," J. Opt. Soc. Am. 66, 751-760 (1976).
    [CrossRef]
  9. S. Y. Shin and L. B. Felsen, "Gaussian beam modes by multipoles with complex source points," J. Opt. Soc. Am. 67, 699-700 (1977).
    [CrossRef]
  10. E. Heyman and L. B. Felsen, "Complex source pulse-beam fields," J. Opt. Soc. Am. A 6, 806-817 (1989).
    [CrossRef]
  11. E. Heyman and L. B. Felsen, "Gaussian beam and pulsed-beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics," J. Opt. Soc. Am. A 18, 1588-1611 (2001).
    [CrossRef]
  12. C. J. R. Sheppard and S. Saghafi, "Beam modes beyond the paraxial approximation: a scalar treatment," Phys. Rev. A 57, 2971-2979 (1998).
    [CrossRef]
  13. C. J. R. Sheppard, "High-aperture beams," J. Opt. Soc. Am. A 18, 1579-1587 (2001).
    [CrossRef]
  14. A. L. Cullen and P. K. Yu, "Complex source-point theory of the electromagnetic open resonator," Proc. R. Soc. London, Ser. A 366, 155-171 (1979).
    [CrossRef]
  15. C. J. R. Sheppard and S. Saghafi, "Electromagnetic Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 16, 1381-1386 (1999).
    [CrossRef]
  16. C. J. R. Sheppard and S. Saghafi, "Electric and magnetic dipole beams beyond the paraxial approximation," Optik 110, 487-491 (1999).
  17. 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]
  18. 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]
  19. C. J. R. Sheppard and K. Larkin, "Optimal concentration of electromagnetic radiation," J. Mod. Opt. 41, 1495-1505 (1994).
    [CrossRef]
  20. P. Carruthers and M. M. Nieto, "Phase and angle variables in quantum mechanics," Rev. Mod. Phys. 40, 411-440 (1968).
    [CrossRef]
  21. T. Opatrný, "Mean value and uncertainty in optical phase--a simple mechanical analogy," J. Phys. A 27, 7201-7208 (1994).
    [CrossRef]
  22. T. Opatrný, "Number-phase uncertainty relations," J. Phys. A 28, 6961-6975 (1995).
    [CrossRef]
  23. It should be noted that the operator Rs could be alternatively derived directly from Eq. asRs=(L+iqez×u)·(L−iqez×u),and on using the commutation relation given in Eq. .

2006 (2)

M. Alonso, R. Borghi, and M. Santarsiero, "Nonparaxial fields with maximum joint spatial-directional localization. I. Scalar case," J. Opt. Soc. Am. A 23, 000-000 (2006).

M. Alonso, R. Borghi, and M. Santarsiero, "Nonparaxial fields with maximum joint spatial-directional localization. II. Vector case," J. Opt. Soc. Am. A 23, 000-000 (2006).
[CrossRef]

2001 (2)

2000 (1)

1999 (3)

1998 (1)

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

1995 (1)

T. Opatrný, "Number-phase uncertainty relations," J. Phys. A 28, 6961-6975 (1995).
[CrossRef]

1994 (2)

T. Opatrný, "Mean value and uncertainty in optical phase--a simple mechanical analogy," J. Phys. A 27, 7201-7208 (1994).
[CrossRef]

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

1989 (1)

1979 (1)

A. L. Cullen and P. K. Yu, "Complex source-point theory of the electromagnetic open resonator," Proc. R. Soc. London, Ser. A 366, 155-171 (1979).
[CrossRef]

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]

1977 (1)

1976 (1)

1971 (2)

J. B. Keller and W. Streifer, "Complex rays with an application to Gaussian beams," J. Opt. Soc. Am. 61, 40-43 (1971).
[CrossRef]

G. A. Deschamps, "Gaussian beams as a bundle of complex rays," Electron. Lett. 7, 684-685 (1971).
[CrossRef]

1969 (1)

1968 (1)

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

1967 (1)

Yu. A. Kravtsov, "Complex ray and complex caustics," Radiophys. Quantum Electron. 10, 719-730 (1967).
[CrossRef]

Alonso, M.

M. Alonso, R. Borghi, and M. Santarsiero, "Nonparaxial fields with maximum joint spatial-directional localization. I. Scalar case," J. Opt. Soc. Am. A 23, 000-000 (2006).

M. Alonso, R. Borghi, and M. Santarsiero, "Nonparaxial fields with maximum joint spatial-directional localization. II. Vector case," J. Opt. Soc. Am. A 23, 000-000 (2006).
[CrossRef]

Alonso, M. A.

Arnaud, J.

Borghi, R.

M. Alonso, R. Borghi, and M. Santarsiero, "Nonparaxial fields with maximum joint spatial-directional localization. II. Vector case," J. Opt. Soc. Am. A 23, 000-000 (2006).
[CrossRef]

M. Alonso, R. Borghi, and M. Santarsiero, "Nonparaxial fields with maximum joint spatial-directional localization. I. Scalar case," J. Opt. Soc. Am. A 23, 000-000 (2006).

Carruthers, P.

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

Cullen, A. L.

A. L. Cullen and P. K. Yu, "Complex source-point theory of the electromagnetic open resonator," Proc. R. Soc. London, Ser. A 366, 155-171 (1979).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, "Gaussian beams as a bundle of complex rays," Electron. Lett. 7, 684-685 (1971).
[CrossRef]

Felsen, L. B.

Forbes, G. W.

Heyman, E.

Keller, J. B.

Kravtsov, Yu. A.

Yu. A. Kravtsov, "Complex ray and complex caustics," Radiophys. Quantum Electron. 10, 719-730 (1967).
[CrossRef]

Larkin, K.

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

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]

Saghafi, S.

C. J. R. Sheppard and S. Saghafi, "Electric and magnetic dipole beams beyond the paraxial approximation," Optik 110, 487-491 (1999).

C. J. R. Sheppard and S. Saghafi, "Electromagnetic Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 16, 1381-1386 (1999).
[CrossRef]

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

Santarsiero, M.

M. Alonso, R. Borghi, and M. Santarsiero, "Nonparaxial fields with maximum joint spatial-directional localization. I. Scalar case," J. Opt. Soc. Am. A 23, 000-000 (2006).

M. Alonso, R. Borghi, and M. Santarsiero, "Nonparaxial fields with maximum joint spatial-directional localization. II. Vector case," J. Opt. Soc. Am. A 23, 000-000 (2006).
[CrossRef]

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, "Electromagnetic Gaussian beams beyond the paraxial approximation," J. Opt. Soc. Am. A 16, 1381-1386 (1999).
[CrossRef]

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 S. Saghafi, "Electric and magnetic dipole beams beyond the paraxial approximation," Optik 110, 487-491 (1999).

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

Shin, S. Y.

Streifer, W.

Yu, P. K.

A. L. Cullen and P. K. Yu, "Complex source-point theory of the electromagnetic open resonator," Proc. R. Soc. London, Ser. A 366, 155-171 (1979).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (1)

G. A. Deschamps, "Gaussian beams as a bundle of complex rays," Electron. Lett. 7, 684-685 (1971).
[CrossRef]

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. 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. Soc. Am. (3)

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

J. Phys. A (2)

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]

Opt. Lett. (1)

Optik (1)

C. J. R. Sheppard and S. Saghafi, "Electric and magnetic dipole beams beyond the paraxial approximation," Optik 110, 487-491 (1999).

Phys. Rev. A (1)

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

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

A. L. Cullen and P. K. Yu, "Complex source-point theory of the electromagnetic open resonator," Proc. R. Soc. London, Ser. A 366, 155-171 (1979).
[CrossRef]

Radiophys. Quantum Electron. (1)

Yu. A. Kravtsov, "Complex ray and complex caustics," Radiophys. Quantum Electron. 10, 719-730 (1967).
[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 (1)

It should be noted that the operator Rs could be alternatively derived directly from Eq. asRs=(L+iqez×u)·(L−iqez×u),and on using the commutation relation given in Eq. .

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Figures (5)

Fig. 1
Fig. 1

Behavior of δ r = arctan Δ r versus Δ θ for the MUFs (solid curve) and for the CFFs (circles) in the (a) scalar and (b) vectorial case.

Fig. 2
Fig. 2

Behavior of the normalized angular spectrum (circles) of (a) scalar and (b) vectorial MUFs, as functions of θ π , for w = 0.7 . Solid curves are best fits obtained by use of Eq. (21) for the scalar case (with q 1.96 ), and Eq. (22) for the vectorial case (with q 1.43 ).

Fig. 3
Fig. 3

Correspondence between the parameters q (corresponding to the CFFs) and w (corresponding to the MUFs) provided by a best fit of the corresponding angular spectra for (a) scalar and (b) vector fields. The solid curves indicate the asymptotic behaviors.

Fig. 4
Fig. 4

Behavior, as a function of q, of Δ θ (solid curve) and Δ ̃ θ (circles) in the (a) scalar and (b) vectorial case.

Fig. 5
Fig. 5

Behavior, as a function of q, of the relative difference (expressed in percent) between Δ θ and Δ ̃ θ (solid curve) and between Δ θ and Δ ̂ θ (dotted curve).

Equations (42)

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U ( r ) = 4 π A ( u ) exp ( i u r ) d Ω ,
Δ θ = arccos [ 4 π A ( u ) 2 cos θ d Ω 4 π A ( u ) 2 d Ω ] ,
Δ r = [ 4 π A * ( u ) L 2 A ( u ) d Ω 4 π A ( u ) 2 d Ω ] 1 2 ,
L 2 = 1 sin θ θ [ sin θ θ ] 1 sin 2 θ 2 ϕ 2 ,
Q s = 2 w 4 ( 1 cos θ ) + L 2 ,
E ( r ) = 4 π A ( u ) exp ( i u r ) d Ω ,
Δ θ = arccos [ 4 π A ( u ) 2 cos θ d Ω 4 π A ( u ) 2 d Ω ] ,
Δ r = [ 4 π A * ( u ) L 2 A ( u ) d Ω 4 π A ( u ) 2 d Ω ] 1 2 .
Q v = 2 w 4 ( 1 cos θ ) + P u L 2 ,
A w ( u ) = [ α U 1 ( u ) + β U 2 ( u ) ] F w ( θ ) ,
U 1 ( u ) = ( 1 + cos θ ) ( cos ϕ e θ sin ϕ e ϕ ) ,
U 2 ( u ) = ( 1 + cos θ ) ( sin ϕ e θ + cos ϕ e ϕ ) ,
A ( u ) A ( u ) exp ( i u r 0 ) .
A q ( u ) = A 0 exp ( q u z ) = A 0 exp ( q cos θ ) .
Δ θ = arccos [ 2 q cosh ( 2 q ) sinh ( 2 q ) 2 q sinh ( 2 q ) ] ,
Δ r = [ 2 q cosh ( 2 q ) sinh ( 2 q ) 2 sinh ( 2 q ) ] 1 2 .
A ( u ) = α U 1 ( u ) + β U 2 ( u ) .
A q ( u ) = exp ( q cos θ ) A ( u ) = exp ( q cos θ ) [ α U 1 ( u ) + β U 2 ( u ) ] .
Δ θ = arccos { ( 3 + 10 q 16 q 2 + 16 q 3 ) exp ( 4 q ) + 3 + 2 q 2 q [ ( 1 4 q + 8 q 2 ) exp ( 4 q ) 1 ] } ,
Δ r = [ 1 + 3 q + ( 1 + q + 4 q 2 + 8 q 3 ) exp ( 4 q ) ( 1 4 q + 8 q 2 ) exp ( 4 q ) 1 ] 1 2 .
A ( θ ) = exp [ q ( cos θ 1 ) ] ,
A ( θ ) = 1 + cos θ 2 exp [ q ( cos θ 1 ) ]
[ u x , L ] = i u z .
Δ L 2 Δ u x 2 u z 2 4 + { u z , L } 2 4 u z 2 4 ,
Δ O 2 = O 2 O 2 ,
O = 2 π A * ( u ) O A ( u ) d θ 2 π A ( u ) 2 d θ .
Δ r tan Δ ̃ θ 1 2 ,
Δ ̃ θ = arctan ( Δ u x u z ) = arctan ( sin 2 θ cos θ ) .
L A q ( u ) = i q u x A q ( u ) .
L A q ( u ) = i q e z × u A q ( u ) .
Δ L 2 Δ e z × u 2 [ L j , { e z × u } j ] 2 4 ,
[ L j , u k ] = i ϵ j k m u m ,
[ L j , { e z × u } j ] = ϵ j z k [ L j , u k ] = i ϵ j z k ϵ j k m u m = 2 i u z .
Δ r 2 sin 2 θ u z 2 = cos θ 2 .
Δ r tan Δ ̃ θ 1 ,
Δ ̃ θ = arctan 2 sinh ( 2 q ) 2 q cosh ( 2 q ) sinh ( 2 q ) .
R s = 2 q ( 1 cos θ ) + q 2 sin 2 θ + L 2 ,
R v = 2 q ( 1 cos θ ) + q 2 sin 2 θ + P u L 2 ,
R v = 4 q ( 1 cos θ ) + q 2 sin 2 θ + P u L 2 ,
sin 2 Δ ̂ θ cos 4 Δ ̂ θ = sin 2 θ cos θ 4 ,
Δ ̂ θ = arccos ( 2 1 + 1 + 4 sin 2 θ cos θ 4 ) .
R s = ( L + i q e z × u ) ( L i q e z × u ) ,

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