Abstract

The Jones calculus is a well known method for analyzing the polarization of a fully polarized beam. It deals with a beam having spatially homogeneous polarization. In recent years, axially symmetric polarized beams, where the polarization is not homogeneous in its cross section, have attracted great interest. In the present article, we show the formula for the rotation of beams and optical elements on the angularly variant term-added Jones calculus, which is required for analyzing axially symmetric beams. In addition, we introduce an extension of the Jones calculus: use of the polar coordinate basis. With this calculus, the representation of some angularly variant beams and optical elements are simplified and become intuitive. We show definitions, examples, and conversion formulas between different notations.

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References

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  1. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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  14. Noted that (cos(ξ+θ)sin(ξ+θ)) is not equivalent to (cosξsinξ). This can be confirmed by substituting θ = π/2; (cos(ξ+π/2)sin(ξ+π/2))=(−sinξcosξ) is not radial polarization, but azimuthal polarization.
  15. G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. 32, 1468–1470 (2007).
    [CrossRef] [PubMed]

2010

2009

2008

2007

2003

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

2000

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

1996

1941

Azzam, R.M.A.

R.M.A. Azzam and N.M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Bashara, N.M.

R.M.A. Azzam and N.M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Bu, J.

Burge, R. E.

Cottrell, D. M.

Davis, J. A.

Dorn, R.

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Gao, B. Z.

Glöckl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Hecht, E.

E. Hecht, “A mathematical description of polarization,” in Optics , 4th ed. (Addison Wesley, 2002), chap. 8.13, pp. 373–379.

Heckenberg, N. R.

Jackel, S.

Jones, R. C.

Kozawa, Y.

Leuchs, G.

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Lumer, Y.

Machavariani, G.

Meir, A.

Moh, K. J.

Moreno, I.

Moshe, I.

Nieminen, T. A.

Quabis, S.

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

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Rubinsztein-Dunlop, H.

Ruiz, I.

Sato, S.

Schadt, M.

Shurcliff, W.A.

W.A. Shurcliff, Polarized Light: Production and Use (Harvard University Press, 1962).

Stalder, M.

Yuan, X.-C.

Zhan, Q.

Adv. Opt. Photon.

Appl. Opt.

J. Opt. Soc. Am.

Opt. Commun.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

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

Other

Noted that (cos(ξ+θ)sin(ξ+θ)) is not equivalent to (cosξsinξ). This can be confirmed by substituting θ = π/2; (cos(ξ+π/2)sin(ξ+π/2))=(−sinξcosξ) is not radial polarization, but azimuthal polarization.

W.A. Shurcliff, Polarized Light: Production and Use (Harvard University Press, 1962).

R.M.A. Azzam and N.M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

E. Hecht, “A mathematical description of polarization,” in Optics , 4th ed. (Addison Wesley, 2002), chap. 8.13, pp. 373–379.

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Figures (3)

Fig. 1
Fig. 1

Schematic representation of the electric field distribution (a snapshot) of axially symmetric beams, (a) radial polarization, and (b) azimuthal polarization. Corresponding Jones vectors are shown in the text. Note that only eight arrows are shown in each plot, but the direction of oscillation varies continuously with polar angle ξ. The background gray color schematically represents a donut-like distribution of beam intensity.

Fig. 2
Fig. 2

Schematic representation of rotation. (a) shows the rotation of the direction of the electric field at its original position. (b) shows the rotation around the origin of the coordinate without changing the direction of the electric field.

Fig. 3
Fig. 3

Schematic representation of (a) an angularly variant half-wave plate (av-HWP) with a fast axis tilted by ξ/2 from the x axis at angle ξ. Thick solid lines indicate the fast axis, and dotted lines indicate the slow axis. (b) Angularly variant linear polarizer (av-LP) with transmission axis radial. Double-headed arrows indicate the transmission axis.

Tables (2)

Tables Icon

Table 1 xy- and Polar-Jones Vectors for Some Polarization States

Tables Icon

Table 2 xy- and Polar-Jones Matrices for Some Optical Elements

Equations (35)

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J radial ( ξ ) = ( cos ξ sin ξ ) ,
J azimuthal ( ξ ) = ( sin ξ cos ξ ) .
J ( θ ) = R ( θ ) J ,
J radial ( θ ) ( ξ ) = R ( θ ) J radial ( ξ ) = ( cos θ sin θ sin θ cos θ ) = ( cos ( ξ + θ ) sin ( ξ + θ ) ) , ( cos ξ sin ξ )
J ( θ ) ( ξ ) = R ( θ ) J ( ξ θ ) ,
J radial ( θ ) = R ( θ ) J radial ( ξ θ ) = ( cos θ sin θ sin θ cos θ ) = ( cos ξ sin ξ ) , ( cos ( ξ θ ) sin ( ξ θ ) )
M av HWP ( ξ ) = ( cos ξ sin ξ sin ξ cos ξ ) .
M av HWP ( ξ ) J horizontal = ( cos ξ sin ξ sin ξ cos ξ ) ( 1 0 ) = ( cos ξ sin ξ ) .
M ( θ ) = R ( θ ) MR ( θ ) .
M ( θ ) ( ξ ) = R ( θ ) M ( ξ θ ) R ( θ ) .
M av HWP ( π / 2 ) ( ξ ) = R ( π / 2 ) M av HWP ( ξ ) R ( π / 2 ) = ( sin ξ cos ξ cos ξ sin ξ ) .
M av HWP ( π / 2 ) ( ξ ) J horizontal = ( sin ξ cos ξ ) ,
M av LP ( ξ ) = ( cos 2 ξ sin ξ cos ξ sin ξ cos ξ sin 2 ξ ) ,
M av LP ( θ ) ( ξ ) = R ( θ ) M av LP ( ξ θ ) R ( θ ) ,
J ̌ = ( E rad ( ξ ) E az ( ξ ) ) .
J ̌ radial = ( 1 0 ) ,
J ̌ azimuthal = ( 0 1 ) .
J ̌ ( ξ ) = R ( ξ ) J ( ξ ) ,
J ( ξ ) = R ( ξ ) J ̌ ( ξ ) .
J ̌ horizontal = R ( ξ ) J horizontal = ( cos ξ sin ξ ) .
M ̌ av HWP = ( cos ξ sin ξ sin ξ cos ξ ) ,
M ̌ av HWP J ̌ horizontal = ( cos ξ sin ξ sin ξ cos ξ ) ( cos ξ sin ξ ) = ( 1 0 ) ,
MJ = R ( ξ ) M ̌ J ̌ ,
MJ = R ( ξ ) M ̌ R ( ξ ) J ,
M = R ( ξ ) M ̌ R ( ξ ) ,
M ̌ = R ( ξ ) MR ( ξ ) ,
M ̌ av LP = ( 1 0 0 0 ) .
J ̌ LCP ( ξ ) = R ( ξ ) J LCP ,
M ̌ av LP ( ξ ) J ̌ LCP ( ξ ) = M ̌ av LP ( ξ ) R ( ξ ) J LCP = 1 2 ( cos ξ + i sin ξ 0 ) .
M av LP J LCP = 1 2 ( cos 2 ξ + i sin ξ cos ξ sin ξ cos ξ + i sin 2 ξ ) .
J ̌ ( θ ) ( ξ ) = J ̌ ( ξ θ ) ,
M ̌ ( θ ) ( ξ ) = M ̌ ( ξ θ ) .
L E ( J ̌ ) = 0 2 π E rad ( ξ ) d ξ ,
L E ( J ̌ radial ) = 0 2 π 1 d ξ = 2 π ,
L E ( M ̌ av LP ( ξ ) J ̌ LCP ( ξ ) ) = 0 2 π 1 2 ( cos ξ + i sin ξ ) d ξ = 0.

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