Abstract

On the basis of a formal analogy with the irradiance moments, analytical definitions are proposed for the width of both the transverse and the longitudinal component of rotationally-symmetric radially-polarized fields at the focal plane of a high-focusing optical system. The beam width of the whole field is also introduced. The transverse beam size is thus associated with the overall spatial structure of the field. The beam-width definitions are applied to an illustrative example, which enables us to show that, at the focal plane, the power contained within a circle whose radius is given by the proposed beam widths represents the main part of the total power.

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  1. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, Cambridge, 2007).
  2. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).
  3. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
    [CrossRef] [PubMed]
  4. D. W. Diehl, R. W. Schoonover, and T. D. Visser, “The structure of focused, radially polarized fields,” Opt. Express 14(7), 3030–3038 (2006).
    [CrossRef] [PubMed]
  5. S. Lavi, R. Prochaska, and E. Keren, “Generalized beam parameters and transformation laws for partially coherent light,” Appl. Opt. 27(17), 3696–3703 (1988).
    [CrossRef] [PubMed]
  6. M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttg.) 82, 173–181 (1989).
  7. A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
    [CrossRef]
  8. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), 1027–1049 (1992).
    [CrossRef]
  9. R. Martínez-Herrero and P. M. Mejías, “Expansion of the cross-spectral density function of general fields and its application to beam characterization,” Opt. Commun. 94(4), 197–202 (1992).
    [CrossRef]
  10. R. Martínez-Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
    [CrossRef]
  11. P. M. Mejías and R. Martínez-Herrero, “Time-resolved spatial parametric characterization of pulsed light beams,” Opt. Lett. 20(7), 660–662 (1995).
    [CrossRef] [PubMed]
  12. J. R., “Martín de Los Santos, R. Martínez-Herrero and P. M. Mejías, “Parametric characterization of Gaussian beams propagating through active media,” Opt. Quantum Electron. 28, 1021–1027 (1996).
  13. R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, Berlin, 2009).
  14. R. Martínez-Herrero, P. M. Mejías, and A. Manjavacas, “On the definition of beam width of highly-focused radially-polarized light fields,” Proc. SPIE 7712, 77122K (2010), doi:.
    [CrossRef]
  15. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1991).

2010 (1)

R. Martínez-Herrero, P. M. Mejías, and A. Manjavacas, “On the definition of beam width of highly-focused radially-polarized light fields,” Proc. SPIE 7712, 77122K (2010), doi:.
[CrossRef]

2006 (1)

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

2001 (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

1996 (1)

J. R., “Martín de Los Santos, R. Martínez-Herrero and P. M. Mejías, “Parametric characterization of Gaussian beams propagating through active media,” Opt. Quantum Electron. 28, 1021–1027 (1996).

1995 (1)

1993 (1)

R. Martínez-Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[CrossRef]

1992 (2)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), 1027–1049 (1992).
[CrossRef]

R. Martínez-Herrero and P. M. Mejías, “Expansion of the cross-spectral density function of general fields and its application to beam characterization,” Opt. Commun. 94(4), 197–202 (1992).
[CrossRef]

1990 (1)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

1989 (1)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttg.) 82, 173–181 (1989).

1988 (1)

Bastiaans, M. J.

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttg.) 82, 173–181 (1989).

Diehl, D. W.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Keren, E.

Lavi, S.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Manjavacas, A.

R. Martínez-Herrero, P. M. Mejías, and A. Manjavacas, “On the definition of beam width of highly-focused radially-polarized light fields,” Proc. SPIE 7712, 77122K (2010), doi:.
[CrossRef]

Martínez-Herrero, R.

R. Martínez-Herrero, P. M. Mejías, and A. Manjavacas, “On the definition of beam width of highly-focused radially-polarized light fields,” Proc. SPIE 7712, 77122K (2010), doi:.
[CrossRef]

P. M. Mejías and R. Martínez-Herrero, “Time-resolved spatial parametric characterization of pulsed light beams,” Opt. Lett. 20(7), 660–662 (1995).
[CrossRef] [PubMed]

R. Martínez-Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[CrossRef]

R. Martínez-Herrero and P. M. Mejías, “Expansion of the cross-spectral density function of general fields and its application to beam characterization,” Opt. Commun. 94(4), 197–202 (1992).
[CrossRef]

Mejías, P. M.

R. Martínez-Herrero, P. M. Mejías, and A. Manjavacas, “On the definition of beam width of highly-focused radially-polarized light fields,” Proc. SPIE 7712, 77122K (2010), doi:.
[CrossRef]

P. M. Mejías and R. Martínez-Herrero, “Time-resolved spatial parametric characterization of pulsed light beams,” Opt. Lett. 20(7), 660–662 (1995).
[CrossRef] [PubMed]

R. Martínez-Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[CrossRef]

R. Martínez-Herrero and P. M. Mejías, “Expansion of the cross-spectral density function of general fields and its application to beam characterization,” Opt. Commun. 94(4), 197–202 (1992).
[CrossRef]

Prochaska, R.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

R., J.

J. R., “Martín de Los Santos, R. Martínez-Herrero and P. M. Mejías, “Parametric characterization of Gaussian beams propagating through active media,” Opt. Quantum Electron. 28, 1021–1027 (1996).

Schoonover, R. W.

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

Visser, T. D.

Weber, H.

R. Martínez-Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[CrossRef]

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), 1027–1049 (1992).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

Opt. Commun. (1)

R. Martínez-Herrero and P. M. Mejías, “Expansion of the cross-spectral density function of general fields and its application to beam characterization,” Opt. Commun. 94(4), 197–202 (1992).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Opt. Quantum Electron. (3)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24(9), 1027–1049 (1992).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, and H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[CrossRef]

J. R., “Martín de Los Santos, R. Martínez-Herrero and P. M. Mejías, “Parametric characterization of Gaussian beams propagating through active media,” Opt. Quantum Electron. 28, 1021–1027 (1996).

Optik (Stuttg.) (1)

M. J. Bastiaans, “Propagation laws for the second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttg.) 82, 173–181 (1989).

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[CrossRef] [PubMed]

Proc. SPIE (2)

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, and A. Manjavacas, “On the definition of beam width of highly-focused radially-polarized light fields,” Proc. SPIE 7712, 77122K (2010), doi:.
[CrossRef]

Other (3)

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1991).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, Cambridge, 2007).

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, Berlin, 2009).

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Figures (4)

Fig. 1
Fig. 1

Illustrating the notation and the geometry of the problem. The value θ 0 corresponds to the semi-aperture angle of the aplanatic system.

Fig. 2
Fig. 2

Irradiance (arbitrary units) at the focal plane versus the radial distance r to the propagation axis z (in the figures, the curves have been represented in terms of the dimensionless parameter kr). The figures refer to the transverse component (Fig. 2.a), the longitudinal component (Fig. 2.b), and the whole field (Fig. 2.c). In all the figures, the same scale has been used for ordinates, and the filling factor f 0 has been chosen equal to 1.

Fig. 3
Fig. 3

Dimensionless parameter 2kw at the focal plane, associated with the transverse component (dashed line, 2 k w T ), the longitudinal component (dotted line, 2 k w L ), and the whole field (continuous line, 2 k w G ) versus the filling factor f 0.

Fig. 4
Fig. 4

Power-content ratio P 2w (defined by Eq. (40)) in terms of f 0. Dashed line, dotted line and continuous line correspond, respectively, to the transverse component, longitudinal component and whole field.

Equations (42)

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< x 2 + y 2 > < r 2 > = r 2 | E | 2 r d r d ϕ I = I x < r 2 > x I + I y < r 2 > y I ,
< r 2 > j r 2 | E j | 2 r d r d ϕ I j ,    j   =   x ,   y ,
I j | E j | 2 r d r d ϕ ,    j   =   x ,   y ,
E ( r , ϕ ) = A 0 θ 0 0 2 π E i ( θ ) cos θ ( cos θ cos φ cos θ sin φ sin θ ) exp [ i k r sin θ cos ( ϕ φ ) ] sin θ d θ d φ ,
ρ sin θ .
( E x E y ) = A 0 a 0 2 π E i ( ρ ) ( 1 ρ 2 ) 1 / 4 ( cos φ sin φ ) exp [ i k r ρ cos ( ϕ φ ) ] ρ d ρ d φ ,
E z = A 0 a 0 2 π ρ E i ( ρ ) ( 1 ρ 2 ) 1 / 4 exp [ i k r ρ cos ( ϕ φ ) ] ρ d ρ d φ ,
E i ( ρ ) = ρ h ( ρ ) ,
E T ( E x ( r , ϕ ) E y ( r , ϕ ) ) = 2 π i A ( cos ϕ sin ϕ ) f ( r ) ,
f ( r ) = 0 a E i ( ρ ) ( 1 ρ 2 ) 1 4 J 1 ( k r ρ ) ρ d ρ ,
0 2 π ( cos φ sin φ ) exp [ i k r ρ cos ( ϕ φ ) ] d φ = 2 π i ( cos ϕ sin ϕ ) J 1 ( k r ρ ) .
I T ( r ) 0 2 π | E T ( r , ϕ ) | 2 d ϕ = ( 2 π ) 3 | A | 2 | f ( r ) | 2 .
f ( r ) = f 1 ( r ) + f 2 ( r ) ,
f 1 ( r ) = a 2 k r F ( a ) J 2 ( k r a ) ,
f 2 ( r ) = 1 k r 0 a ρ 2 J 2 ( k r ρ ) F ( ρ ) ρ d ρ ,
F ( ρ ) = h ( ρ ) ( 1 ρ 2 ) 1 4 .
w T 2 w T 1 2 I 1 + w T 2 2 I 2 I T ,
I T 0 0 2 π | E T ( r , ϕ ) | 2 r d r d ϕ = 0 I T ( r ) r d r = 8 π 3 k 2 | A | 2 0 a | ρ h ( ρ ) ( 1 ρ 2 ) 1 4 | 2 ρ d ρ ,
I j = ( 2 π ) 3 | A | 2 0 | f j ( r ) | 2 r d r ,    j   =  1 , 2 .
w T 2 2 = ( 2 π ) 3 | A | 2 0 r 2 | f 2 ( r ) | 2 r d r I 2 = ( 2 π ) 3 | A | 2 k 4 I 2 0 a ρ 2 | F ( ρ ) ρ | 2 ρ d ρ .
| f 1 ( r ) | 2 = a 6 | F ( a ) | 2 [ J 2 ( k r a ) k r a ] 2 .
I 1 w T 1 2 = ( 2 π ) 3 | A | 2 4 k 4 a 2 c 2 2 | F ( a ) | 2 ,
0 | f 1 ( r ) | 2 r d r = a 6 | F ( a ) | 2 0 [ J 2 ( k r a ) k r a ] 2 r d r = a 4 4 k 2 | F ( a ) | 2 .
w T 2 = 1 k 2 I 0 T [ a 2 c 2 2 4 | F ( a ) | 2 + 0 a ρ 2 | F ( ρ ) ρ | 2 ρ d ρ ] ,
I 0 T = 0 a | ρ h ( ρ ) ( 1 ρ 2 ) 1 4 | 2 ρ d ρ .
w T 2 = h 0 2 k 2 a 6 c 2 2 16 | F ( a ) | 2 .
E i ( ρ ) = ρ h ( ρ ) = h 0 ρ ( 1 ρ 2 ) 1 4 .
E z ( r , ϕ ) = 2 π A 0 a G ( ρ ) J 0 ( k r ρ ) ρ d ρ ,
G ( ρ ) = ρ 2 h ( ρ ) ( 1 ρ 2 ) 1 4 .
w L 2 = 1 k 2 I 0 L [ c 1 2 2 | G ( a ) | 2 + 0 a | G ( ρ ) ρ | 2 ρ d ρ ] ,
I 0 L = 0 a | ρ 2 h ( ρ ) ( 1 ρ 2 ) 1 4 | 2 ρ d ρ .
w G 2 = I T w T 2 + I L w L 2 I ,
I L 0 0 2 π | E z ( r , ϕ ) | 2 r d r d ϕ ,
w G 2 = 1 k 2 I 0 G { a 2 c 2 2 4 | F ( a ) | 2 + c 1 2 2 | G ( a ) | 2 + 0 a [ ρ 2 | F ( ρ ) ρ | 2 + | G ( ρ ) ρ | 2 ] ρ d ρ } ,
P w j = 0 w j I j ( r ) r d r 0 I j ( r ) r d r ,   j = T , L , G ,
0 r 2 I ( r ) r d r R 0 r 2 I ( r ) r d r R 0 2 R 0 I ( r ) r d r ,
0 R 0 I ( r ) r d r 0 I ( r ) r d r = 1 R 0 I ( r ) r d r 0 I ( r ) r d r 1 1 R 0 2 0 r 2 I ( r ) r d r 0 I ( r ) r d r = 1 < r 2 > R 0 2 .
0 R 0 I ( r ) r d r 0 I ( r ) r d r 1 < r 2 > R 0 2 > 3 4 ,
R 0 2 < r 2 > .
P 2 w j = 0 2 w j I j ( r ) r d r 0 I j ( r ) r d r ,    j   =   T ,   L ,   G .
E i ( ρ ) = ρ h ( ρ ) = ρ exp ( f 2 ρ 2 ω 0 2 ) = ρ exp ( ρ 2 f 0 2 a 2 ) ,
P 2 w j = 0 2 w j I j ( r ) r d r 0 I j ( r ) r d r ,    j   =   T ,   L ,   G .

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