Abstract

The 3×1 generalized Jones vectors (GJVs) [ExEyEz]t (t indicates the transpose) that describe the linear, circular, and elliptical polarization states of an arbitrary three-dimensional (3-D) monochromatic light field are determined in terms of the geometrical parameters of the 3-D vibration of the time-harmonic electric field. In three dimensions, there are as many distinct linear polarization states as there are points on the surface of a hemisphere, and the number of distinct 3-D circular polarization states equals that of all two-dimensional (2-D) polarization states on the Poincaré sphere, of which only two are circular states. The subset of 3-D polarization states that results from the superposition of three mutually orthogonal x, y, and z field components of equal amplitude is considered as a function of their relative phases. Interesting contours of equal ellipticity and equal inclination of the normal to the polarization ellipse with respect to the x axis are obtained in 2-D phase space. Finally, the 3×3 generalized Jones calculus, in which elastic scattering (e.g., by a nano-object in the near field) is characterized by the 3-D linear transformation Es=TEi, is briefly introduced. In such a matrix transformation, Ei and Es are the 3×1 GJVs of the incident and scattered waves and T is the 3×3 generalized Jones matrix of the scatterer at a given frequency and for given directions of incidence and scattering.

© 2011 Optical Society of America

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References

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  2. H. Poincaré, Théorie Mathématique de la Lumière (Gauthiers-Villares, 1892).
  3. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987), Sec. 1.3.
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    [CrossRef]
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    [CrossRef]
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  10. A. T. Adams and E. Mendelovich, “The near-field polarization ellipse,” IEEE Trans. Antennas Propag. AP-21, 124–126(1973).
    [CrossRef]
  11. C. Brosseau and A. Dogariu, “Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield,” in Vol.  49 of Progress in Optics, E.Wolf, ed. (Elsevier, 2006), Chap. 4.
    [CrossRef]
  12. A. Norrman, T. Setälä, and A. T. Friberg, “Partial coherence and partial polarization in random evanescent fields on lossless interfaces,” J. Opt. Soc. Am. A 28, 391–400 (2011).
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  13. R. C. Dorf and J. A. Svoboda, Introduction to Electric Circuits, 4th ed. (Wiley, 1999), Sec. 12.3.
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    [CrossRef]
  15. R. M. A. Azzam, “Phase shifts that accompany total internal reflection at a dielectric–dielectric interface,” J. Opt. Soc. Am. A 21, 1559–1563 (2004).
    [CrossRef]
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    [CrossRef]

2011 (2)

2010 (1)

2009 (1)

2007 (1)

2006 (2)

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

C. Brosseau and A. Dogariu, “Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield,” in Vol.  49 of Progress in Optics, E.Wolf, ed. (Elsevier, 2006), Chap. 4.
[CrossRef]

2004 (1)

2001 (1)

F. de Fornel, Evanescent Waves (Springer, 2001).

1999 (1)

R. C. Dorf and J. A. Svoboda, Introduction to Electric Circuits, 4th ed. (Wiley, 1999), Sec. 12.3.

1995 (1)

J. T. Verdeyen, Laser Electronics (Prentice Hall, 1995), Sec. 3.2.

1987 (1)

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987), Sec. 1.3.

1984 (1)

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, 1984), Sec. 6.3.

1973 (1)

A. T. Adams and E. Mendelovich, “The near-field polarization ellipse,” IEEE Trans. Antennas Propag. AP-21, 124–126(1973).
[CrossRef]

1969 (1)

R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
[CrossRef]

1892 (1)

H. Poincaré, Théorie Mathématique de la Lumière (Gauthiers-Villares, 1892).

Adams, A. T.

A. T. Adams and E. Mendelovich, “The near-field polarization ellipse,” IEEE Trans. Antennas Propag. AP-21, 124–126(1973).
[CrossRef]

Azzam, R. M. A.

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987), Sec. 1.3.

Brosseau, C.

C. Brosseau and A. Dogariu, “Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield,” in Vol.  49 of Progress in Optics, E.Wolf, ed. (Elsevier, 2006), Chap. 4.
[CrossRef]

Chen, W.

W. Chen and Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun. 284, 52–56 (2011).
[CrossRef]

de Fornel, F.

F. de Fornel, Evanescent Waves (Springer, 2001).

Dogariu, A.

C. Brosseau and A. Dogariu, “Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield,” in Vol.  49 of Progress in Optics, E.Wolf, ed. (Elsevier, 2006), Chap. 4.
[CrossRef]

Dorf, R. C.

R. C. Dorf and J. A. Svoboda, Introduction to Electric Circuits, 4th ed. (Wiley, 1999), Sec. 12.3.

Fiutowski, J.

Friberg, A. T.

Gu, M.

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, 1984), Sec. 6.3.

Hecht, B.

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

Jia, B.

Józefowski, L.

Kang, H.

Kawalec, T.

Mendelovich, E.

A. T. Adams and E. Mendelovich, “The near-field polarization ellipse,” IEEE Trans. Antennas Propag. AP-21, 124–126(1973).
[CrossRef]

Muller, R. H.

R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
[CrossRef]

Norrman, A.

Novotny, L.

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

Poincaré, H.

H. Poincaré, Théorie Mathématique de la Lumière (Gauthiers-Villares, 1892).

Rubahn, H.-G.

Setälä, T.

Svoboda, J. A.

R. C. Dorf and J. A. Svoboda, Introduction to Electric Circuits, 4th ed. (Wiley, 1999), Sec. 12.3.

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics (Prentice Hall, 1995), Sec. 3.2.

Zhan, Q.

W. Chen and Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun. 284, 52–56 (2011).
[CrossRef]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009).
[CrossRef]

Adv. Opt. Photon. (1)

IEEE Trans. Antennas Propag. (1)

A. T. Adams and E. Mendelovich, “The near-field polarization ellipse,” IEEE Trans. Antennas Propag. AP-21, 124–126(1973).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

W. Chen and Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun. 284, 52–56 (2011).
[CrossRef]

Opt. Express (1)

Surf. Sci. (1)

R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
[CrossRef]

Other (8)

R. C. Dorf and J. A. Svoboda, Introduction to Electric Circuits, 4th ed. (Wiley, 1999), Sec. 12.3.

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

C. Brosseau and A. Dogariu, “Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield,” in Vol.  49 of Progress in Optics, E.Wolf, ed. (Elsevier, 2006), Chap. 4.
[CrossRef]

H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall, 1984), Sec. 6.3.

F. de Fornel, Evanescent Waves (Springer, 2001).

J. T. Verdeyen, Laser Electronics (Prentice Hall, 1995), Sec. 3.2.

H. Poincaré, Théorie Mathématique de la Lumière (Gauthiers-Villares, 1892).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987), Sec. 1.3.

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Figures (6)

Fig. 1
Fig. 1

Right-handed x y z Cartesian coordinate system and the unit sphere of radius r = 1 . A LP state is specified by a radial line through the origin with the zenith and azimuth angles ( θ , ϕ ) of spherical coordinates. A CP state is represented by a great circle with a normal unit vector z ^ = n ^ ( θ , ϕ ) and sense of rotation related to n ^ by the right-hand rule. An EP state is specified by the unit vector z ^ = n ^ ( θ , ϕ ) normal to the x y orbital plane and the orientation and ellipticity angles ψ, ε of the ellipse (not shown) in that plane.

Fig. 2
Fig. 2

Contours of equal ellipticity ( e = constant ) in the δ y δ z phase space for 3-D polarization states generated by the superposition of three mutually orthogonal linear components of equal amplitude [Eqs. (9)]. This family of contours corresponds to values of 2 C 2 [Eq. (15)] in the range from 0.5 to 4.5.

Fig. 3
Fig. 3

Continuation of the e = constant contours of Fig. 2 for values of 2 C 2 [Eq. (15)] from 0 to 0.5. This family of contours represents near-circular polarization states with 0.707 e 1 . The “bull’s eyes” in the triangular regions represent eight distinct 3-D CP states ( e = 1 ).

Fig. 4
Fig. 4

Balanced (zero-sum) set of three phasors E x , E y , and E z [Eqs. (A2)] that generate a CP state with unit normal n ^ = ( 1 , 1 , 1 ) / 3 .

Fig. 5
Fig. 5

Contours of constant inclination angle α (between the normal to the plane of the polarization ellipse and the positive x axis) in the δ y δ z plane for values of cos α from 0.8 to 0.8 in steps of 0.1. (See text for detail.)

Fig. 6
Fig. 6

Geometry of TIR at an angle ϕ at a dielectric–dielectric interface, the reference x y z coordinate system, and the p and s linear polarizations of incident and reflected light.

Equations (27)

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E LP = [ E x E y E z ] t = n ^ = [ sin θ cos ϕ sin θ sin ϕ cos θ ] t .
x ^ = ( sin ϕ ) x ^ ( cos ϕ ) y ^ .
y ^ = ( cos θ cos ϕ ) x ^ + ( cos θ sin ϕ ) y ^ ( sin θ ) z ^ .
E CP = x ^ j y ^ .
E CP = [ E x E y E z ] t = [ ( sin ϕ j cos θ cos ϕ ) ( cos ϕ + j cos θ sin ϕ ) j sin θ ] t .
E EP = ( cos ψ cos ε + j sin ψ sin ε ) x ^ + ( sin ψ cos ε j cos ψ sin ε ) y ^ .
E EP = [ E x E y E z ] t ; E x = cos ε ( sin ϕ cos ψ + cos θ cos ϕ sin ψ ) + j sin ε ( sin ϕ sin ψ cos θ cos ϕ cos ψ ) , E y = cos ε ( cos ϕ cos ψ cos θ sin ϕ sin ψ ) j sin ε ( cos ϕ sin ψ + cos θ sin ϕ cos ψ ) , E z = sin θ ( cos ε sin ψ j sin ε cos ψ ) .
E x = sin ϕ cos ψ + cos θ cos ϕ sin ψ , E y = cos ϕ cos ψ + cos θ sin ϕ sin ψ , E z = sin θ sin ψ .
E x t = cos ( ω t ) , E y t = cos ( ω t + δ y ) , E z t = cos ( ω t + δ z ) .
| E | 2 = 3 2 + A cos ( 2 ω t ) + B sin ( 2 ω t ) ,
A = 1 2 [ 1 + cos ( 2 δ y ) + cos ( 2 δ z ) ] , B = 1 2 [ sin ( 2 δ y ) + sin ( 2 δ z ) ] .
| E | 2 = 1.5 + C cos ( 2 ω t γ ) , C = A 2 + B 2 , γ = arctan ( B / A ) .
| E | max 2 = 1.5 + C , | E | min 2 = 1.5 C .
e = | E | min / | E | max = [ ( 1.5 C ) / ( 1.5 + C ) ] 1 / 2 .
2 C 2 = 1.5 + cos ( 2 δ y ) + cos ( 2 δ z ) + cos ( 2 δ y 2 δ z ) .
E 1 = ( 1 , cos δ y , cos δ z ) , E 2 = ( 0 , sin δ y , sin δ z ) .
n ^ = ( E 1 × E 2 ) / | E 1 × E 2 | .
n ^ = sin ( δ y δ z ) x ^ + ( sin δ z ) y ^ ( sin δ y ) z ^ + [ sin 2 δ y + sin 2 δ z + sin 2 ( δ y δ z ) ] 1 / 2 .
cos α = n ^ . x ^ = sin ( δ y δ z ) [ sin 2 δ y + sin 2 δ z + sin 2 ( δ y δ z ) ] 1 / 2 .
cos α = cos δ y / 0.5 + cos 2 δ y .
E s = T E i ,
E i = [ E i x E i y E i z ] t = [ ( sin ϕ ) E i p E i s ( cos ϕ ) E i p ] t ,
E e = [ E e x E e y E e z ] t .
T 11 = N 2 ( 1 + r p ) , T 22 = ( 1 + r s ) , T 33 = ( 1 r p ) .
r p = exp ( j δ p ) , r s = exp ( j δ s ) .
E x = 2 / 3 exp ( j π / 6 ) , E y = 2 / 3 exp ( j 5 π / 6 ) , E z = 2 / 3 exp ( j π / 2 ) .
E x = 1 , E y = exp ( j 2 π / 3 ) , E z = exp ( j 2 π / 3 ) ,

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