Abstract

We show that the depolarization caused when light is focused with a high-numerical-perture lens is accompanied by a space-variant geometrical phase. This phase results in the formation of modes with helicities and phase singularities that differ from those of the original beam. We show that this effect can be explained as a transverse shift of the rays, which is reminiscent of the recently discovered optical Hall–Magnus effect. Our results show that the asymmetric focal spot associated with the focus of linearly polarized light can be explained through geometrical effects.

© 2007 Optical Society of America

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References

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  1. L. Allen, M. J. Padgett, and M. Babiker, in Progress in Optics, E. Wolf, ed. (Elsevier, 1999), Vol. XXXIX, pp. 291.
    [CrossRef]
  2. M. Onada, S. Murukami, and N. Nagaosa, Phys. Rev. Lett. 93, 083901 (2004).
    [CrossRef]
  3. K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. E 70, 026605 (2004).
    [CrossRef]
  4. M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).
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    [CrossRef]
  6. D. Ganic, X. Gan, and M. Gu, Opt. Express 11, 2747 (2003).
    [CrossRef] [PubMed]
  7. Z. Bomzon, M. Gu, and J. Shamir, Appl. Phys. Lett. 89, 241104 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  10. M. R. Dennis, Opt. Commun. 213, 201 (2002).
    [CrossRef]
  11. A. S. Van de Nes, S. F. Pereira, and J. J. M. Braat, J. Mod. Opt. 53, 677 (2006).
    [CrossRef]

2006 (2)

Z. Bomzon, M. Gu, and J. Shamir, Appl. Phys. Lett. 89, 241104 (2006).
[CrossRef]

A. S. Van de Nes, S. F. Pereira, and J. J. M. Braat, J. Mod. Opt. 53, 677 (2006).
[CrossRef]

2004 (2)

M. Onada, S. Murukami, and N. Nagaosa, Phys. Rev. Lett. 93, 083901 (2004).
[CrossRef]

K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. E 70, 026605 (2004).
[CrossRef]

2003 (1)

2002 (2)

S. M. Barnett, J. Opt. B: Quantum Semiclassical Opt. 4, S7 (2002).
[CrossRef]

M. R. Dennis, Opt. Commun. 213, 201 (2002).
[CrossRef]

2001 (1)

1959 (1)

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

Appl. Phys. Lett. (1)

Z. Bomzon, M. Gu, and J. Shamir, Appl. Phys. Lett. 89, 241104 (2006).
[CrossRef]

J. Mod. Opt. (1)

A. S. Van de Nes, S. F. Pereira, and J. J. M. Braat, J. Mod. Opt. 53, 677 (2006).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (1)

S. M. Barnett, J. Opt. B: Quantum Semiclassical Opt. 4, S7 (2002).
[CrossRef]

Opt. Commun. (1)

M. R. Dennis, Opt. Commun. 213, 201 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. E (1)

K. Y. Bliokh and Y. P. Bliokh, Phys. Rev. E 70, 026605 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

M. Onada, S. Murukami, and N. Nagaosa, Phys. Rev. Lett. 93, 083901 (2004).
[CrossRef]

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

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

Other (2)

L. Allen, M. J. Padgett, and M. Babiker, in Progress in Optics, E. Wolf, ed. (Elsevier, 1999), Vol. XXXIX, pp. 291.
[CrossRef]

M. Gu, Advanced Optical Imaging Theory (Springer-Verlag, 2000).

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Figures (3)

Fig. 1
Fig. 1

Electric energy density associated with the left-hand polarized component (first row), the right-hand polarized component (second row), and the axial component (third row) within the focal spots of a 0.9 NA lens focusing left-hand polarized beams with different topological charges. Note that the left-hand polarized component typically contains about 100 times more energy than the right-hand polarized component and about 10 times more energy than the axial component.

Fig. 2
Fig. 2

Color maps showing 2 χ ( r ) = arg ( E E ) within the focal spots of a 0.9 NA lens focusing beams with left-hand circular, right-hand circular, and linear polarizations (rows) and vorticities m = 0 , 1 , 2 (columns).

Fig. 3
Fig. 3

Illustration showing how the transverse shift of optical rays leads to the formation of modes with different topological charges. A left-hand polarized beam (inner, red rays) is incident on the lens. The rays are bent by the lens, removing a degeneracy associated with polarization, and two sets of rays with orthogonal helicity emerge. The rays with right-hand polarization (outer, blue) undergo a transverse shift ( Δ L ) . The latter rays do not converge at the focal spot, and the shift is manifested as an aberration ( Δ φ = 2 φ ) , which leads to the formation of a mode with topological charge m + 2 .

Equations (8)

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E ( ρ , φ , z ) = ( α β ) E 0 ( ρ ) e i m φ e i k z ,
[ e x e y e z ] = i m { i A [ I m + 0.5 ( I m + 2 e i 2 φ + I m 2 e i 2 φ ) ] A [ 0.5 ( I m + 2 e i 2 φ I m 2 e i 2 φ ) ] A [ I m + 1 e i φ I m 1 e i φ ] } e i m φ .
I m ( u , v ) = 0 γ cos 1 2 θ sin θ ( 1 + cos θ ) J m ( v sin θ sin γ ) exp ( i u cos θ sin 2 γ ) d θ ,
I m ± 2 ( u , v ) = 0 γ cos 1 2 θ sin θ ( 1 cos θ ) J m ± 2 ( v sin θ sin γ ) exp ( i u cos θ sin 2 γ ) d θ ,
I m ± 1 ( u , v ) = 0 γ cos 1 2 θ sin 2 θ J m ± 1 ( v sin θ sin γ ) exp ( i u cos θ sin 2 γ ) d θ .
[ e x e y e z ] = { i A [ I 0 + I 2 cos 2 φ ] i A I 2 sin 2 φ 2 A I 1 cos φ } ,
E = i m + 1 A ( [ 1 i 0 ] I m e i m φ + [ 1 i 0 ] I m + 2 e i ( m + 2 ) φ 2 i [ 0 0 1 ] I m + 1 e i ( m + 1 ) φ ) .
ϕ p = arg E ( r , 0 ) , E ( r , ϕ ) = φ tan 1 [ ( I 0 2 I 2 2 ) sin φ ( I 0 2 + I 2 2 ) cos φ + 2 I 1 2 ] .

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