Abstract

A method to evaluate the physical realizability of an arbitrary three-dimensional vectorial field distribution in the focal area is proposed. A parameter that measures the similarity between the designed (target) field and the physically achievable beam is provided. This analysis is carried out within the framework of the closest electromagnetic field to a given vectorial function, and the procedure is applied to two illustrative cases.

© 2013 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  15. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006), p. 61.

2012 (1)

2010 (4)

2008 (2)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photonics 2, 501 (2008).
[CrossRef]

G. M. Lerman and U. Levy, Opt. Express 16, 4567 (2008).
[CrossRef]

2007 (1)

2004 (2)

2003 (1)

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

2001 (1)

1959 (1)

B. Richards, E. Wolf, B. Richards, and E. Wolf, Proc. R. Soc. A 253, 358 (1959).
[CrossRef]

Bokor, N.

Bosch, S.

Carnicer, A.

Chen, H.

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photonics 2, 501 (2008).
[CrossRef]

Choudhury, A.

Dainty, C.

Davidson, N.

Ding, J.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Hao, X.

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006), p. 61.

Kenny, F.

Khonina, S. N.

Kitamura, K.

Kozawa, Y.

Kuang, C.

Lara, D.

Lerman, G. M.

Leuchs, G.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Levy, U.

Liu, X.

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photonics 2, 501 (2008).
[CrossRef]

Martínez-Herrero, R.

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, J. Opt. Soc. Am. A 18, 1678 (2001).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009), pp. 151–164.

Mejías, P. M.

R. Martínez-Herrero, P. M. Mejías, S. Bosch, and A. Carnicer, J. Opt. Soc. Am. A 18, 1678 (2001).
[CrossRef]

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009), pp. 151–164.

Noda, S.

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006), p. 61.

Piquero, G.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009), pp. 151–164.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Richards, B.

B. Richards, E. Wolf, B. Richards, and E. Wolf, Proc. R. Soc. A 253, 358 (1959).
[CrossRef]

B. Richards, E. Wolf, B. Richards, and E. Wolf, Proc. R. Soc. A 253, 358 (1959).
[CrossRef]

Rodríguez-Herrera, O.

Sakai, K.

Sato, S.

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photonics 2, 501 (2008).
[CrossRef]

Sheppard, C. J.

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photonics 2, 501 (2008).
[CrossRef]

Volotovsky, S. G.

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photonics 2, 501 (2008).
[CrossRef]

Wang, H.-T.

Wang, T.

Wolf, E.

B. Richards, E. Wolf, B. Richards, and E. Wolf, Proc. R. Soc. A 253, 358 (1959).
[CrossRef]

B. Richards, E. Wolf, B. Richards, and E. Wolf, Proc. R. Soc. A 253, 358 (1959).
[CrossRef]

Zhang, B.-F.

Zheng, Z.

Appl. Opt. (1)

J. Opt. Soc. Am. A (3)

Nat. Photonics (1)

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photonics 2, 501 (2008).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

R. Dorn, S. Quabis, and G. Leuchs, Phys. Rev. Lett. 91, 233901 (2003).
[CrossRef]

Proc. R. Soc. A (1)

B. Richards, E. Wolf, B. Richards, and E. Wolf, Proc. R. Soc. A 253, 358 (1959).
[CrossRef]

Other (2)

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006), p. 61.

R. Martínez-Herrero, P. M. Mejías, and G. Piquero, Characterization of Partially Polarized Light Fields (Springer, 2009), pp. 151–164.

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Figures (4)

Fig. 1.
Fig. 1.

Notation and geometry of the problem.

Fig. 2.
Fig. 2.

Projection of F0 on the subspace S.

Fig. 3.
Fig. 3.

Coefficients ρ and ρ0 versus the semi-aperture angle θ0 (in radians) for a longitudinal-only polarized beam. Two functions g(θ) are considered: (i) a Gaussian–Bessel beam and (ii) an apodized Gaussian–Bessel beam with θmin=0.95θ0 and θmax=θ0. In both cases, β=0.5.

Fig. 4.
Fig. 4.

Coefficients ρ and ρ0 versus the semi-aperture angle θ0 (in radians) for a transverse-only radially polarized beam. Functions g(θ) are the same as in Fig. 3.

Equations (21)

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E(r,ϕ,z)=A0θ002πE0(θ,φ)eiksinθrcos(ϕφ)eikcosθzsinθdθdφ,
E0(θ,φ)=P(θ)(f1(θ,φ)e1(φ)+f2(θ,φ)e2(θ,φ)).
e1(φ)=(sinφ,cosφ,0),
e2(θ,φ)=(cosθcosφ,cosθsinφ,sinθ).
F(r,ϕ,0)=A0θ002πF0(θ,φ)eiksinθrcos(ϕφ)sinθdθdφ.
V=(F0·e1)e1+(F0·e2)e2,
inf{|F0G|,GS}=|F0V|,
EF(r,ϕ,0)=A0θ002πV(θ,φ)eiksinθrcos(ϕφ)sinθdθdφ.
ρ=002π|EF(r,ϕ,0)||F(r,ϕ,0)|rdrdϕIFIE,
IE=002π|EF(r,ϕ,0)|2rdrdϕ,
IF=002π|F(r,ϕ,0)|2rdrdϕ.
ρ0=|0θ002π(V·F0)sinθdθdφ|0θ002π|V|2sinθdθdφ0θ002π|F0|2sinθdθdφ.
V(θ,φ)=P(θ)sinθg(θ,φ)e2(θ,φ)
EF(r,ϕ,0)=A0θ002πP(θ)sinθg(θ,φ)e2(θ,φ)eikrsinθcos(ϕφ)sinθdθdφ.
ρ02=0θ002π|P(θ)g(θ,φ)|2sin3θdθdφ0θ002π|P(θ)g(θ,φ)|2sinθdθdφ.
EF(r,ϕ,0)=A0θ002πcosθsinθg(θ)e2(θ,φ)eikrsinθcos(ϕφ)sinθdθdφ.
ρ02=0θ002πcosθ|g(θ)|2sin3θdθdφ0θ002πcosθ|g(θ)|2sinθdθdφ.
limθπ/2V(θ,φ)=F0,
g(θ)=exp[β(sinθsinθ0)2]J1(2βsinθsinθ0),
V(θ,φ)=cosθcosθg(θ)e2(θ,φ).
ρ02=0θ002πcos3θ|g(θ)|2sinθdθdφ0θ002πcosθ|g(θ)|2sinθdθdφ.

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