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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 13 — Jun. 20, 2011
  • pp: 12815–12824
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Matrix calculus for axially symmetric polarized beam

Shigeki Matsuo  »View Author Affiliations


Optics Express, Vol. 19, Issue 13, pp. 12815-12824 (2011)
http://dx.doi.org/10.1364/OE.19.012815


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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.

© 2011 OSA

1. Introduction

Polarization plays an important role in optics. A few methods for representing and calculating polarization have been established. One such method is the Jones calculus [1

1. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–493 (1941). [CrossRef]

4

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

]. In the Jones calculus, the electric field of a polarized beam propagating in the z direction is resolved into x and y components, and is represented as a column vector (the symbol J is frequently used). As well, an optical element is represented as a 2×2 matrix (the symbol M is frequently used). The Jones calculus is a well known method for analyzing a completely polarized beam whose cross section is homogeneously polarized.

Recently, axially symmetric polarized beams (or cylindrical vector beams) have been intensively investigated [5

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

, 6

6. Focus Issue: Unconventional Polarization States of Light, Opt. Express 18(10), 10775–10923 (2010). [PubMed]

]. In an axially symmetric beam, the polarization of the beam is not homogeneous in its cross section, but rather depends on the angle with axial symmetry. Figure 1 schematically represents the electric field distribution of two representative axially symmetric beams: (a) radial polarization and (b) azimuthal polarization. Interest in axially symmetric beams is expanding to include the longitudinal component of an electric field in focus [7

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

], a smaller focus when radial polarization was tightly focused [7

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

, 8

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

], and efficient laser trapping [9

9. T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33, 122–124 (2008). [CrossRef] [PubMed]

, 10

10. Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18, 10828–10833 (2010). [CrossRef] [PubMed]

], to mention few.

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.

Few authors have used Jones vectors and matrices to which were added the angularly variant term [11

11. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 1948–1950 (1996). [CrossRef] [PubMed]

13

13. I. Moreno, J. A. Davis, I. Ruiz, and D. M. Cottrell, “Decomposition of radially and azimuthally polarized beams using a circular-polarization and vortex-sensing diffraction grating,” Opt. Express 18, 7173–7183 (2010). [CrossRef] [PubMed]

], i.e., those incorporating a polar angle (ξ in the present article). For example, the radial polarization shown in Fig. 1(a) is represented as
Jradial(ξ)=(cosξsinξ),
(1)
and the azimuthal polarization shown in Fig. 1(b) is represented as
Jazimuthal(ξ)=(sinξcosξ).
(2)
Even though the angularly variant term was added, the vectors and matrices in this notation are represented with x and y basis, and hence are compatible with those in the conventional (no angle dependence) Jones calculus. That is, the notations can be co-used in the same equation. Consequently, the authors of those reports did not make special remarks on this notation. However, here we identify an important difference between the conventional and angularly variant Jones calculus: the formulas for the rotation of vectors and matrices are different. A key advantage of the Jones calculus is the simplification of the calculation procedure for a rotated optical element. The formulation of rotation is thus important to take full advantage of the Jones calculus.

In the present article, we show that the conventional formula of rotation is not applicable to the angularly variant term-added Jones calculus, and we deduce the correct formula (Sec. 2). Next, we propose a calculus that uses a polar coordinate basis; that is, the electric field of the beam as well as the properties of optical elements are resolved into radial and azimuthal components (Sec. 3). The latter calculus offers an intuitive understanding of the polarization state of an axially symmetric beam. Hereafter, this is referred to as the “polar-Jones” calculus. In order to distinguish the polar-Jones calculus within the Jones calculus, we refer to the Jones calculus based on Cartesian coordinates, rather than polar coordinates, as the “xy-Jones” calculus, regardless of the presence of an angularly variant term. When necessary, the term “av-xy-Jones” calculus is used for emphasizing the presence of an angularly variant term. We also show a simple method for evaluating the presence of the longitudinal component in focus using the polar-Jones representation.

The target of the present article is a collimated and completely symmetrically polarized beam. In addition, we assume that the optical property of an angularly variant optical element varies smoothly with the angle, whereas most of the real optical elements are segmented ones that have stepwise variation of the optical properties. Also, we assume that the oscillation phase of the electric field is kzωt, not ωtkz.

2. Rotation formulas for av-xy-Jones vectors and matrices

2.1. Rotation formula for av-xy-Jones vectors

For a conventional Jones vector, the formula of rotation is given as
J(θ)=R(θ)J,
(3)
where R(θ)=(cosθsinθsinθcosθ) is the rotation matrix. This is not valid for an av-xy-Jones vector. Let us examine an example. If the formula in Eq. (3) was applied to the Jones vector radial polarization J radial(ξ) represented by Eq. (1),
Jradial(θ)(ξ)=R(θ)Jradial(ξ)=(cosθsinθsinθcosθ)=(cos(ξ+θ)sin(ξ+θ)),(cosξsinξ)
(4)
is obtained. Because of the rotational symmetry of the radial polarization, this must be equal to Eq. (1) for any θ, but this is not the case [14

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.

]. This inconsistency can be explained as follows. Equation (3) rotates the polarization at its original position, as illustrated in Fig. 2(a). The correct formula must be a combination of this and another type of rotation: rotation around the origin without changing its direction, as illustrated in Fig. 2(b). Mathematically, the ξ in J must be replaced by ξθ. As a result,
J(θ)(ξ)=R(θ)J(ξθ),
(5)
is the correct formula of rotation for an av-xy-Jones vector. If we apply this formula to the radial polarization,
Jradial(θ)=R(θ)Jradial(ξθ)=(cosθsinθsinθcosθ)=(cosξsinξ),(cos(ξθ)sin(ξθ))
(6)
is obtained. This is the Jones vector before rotation, J radial(ξ). This calculation indicates the validity of the rotation formula Eq. (5). Note that ξ and θ have different roles here; ξ is the coordinate (polar angle) and θ indicates the degree of rotation.

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.

2.2. Rotation formula for av-xy-Jones matrices

An angularly variant half-wave plate (av-HWP) with a fast axis tilted by ξ/2 from the x axis at angle ξ, shown in Fig. 3(a), has been used to generate an axially symmetric beam from a linearly polarized beam [8

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

, 15

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]

]. This is one of the most important angularly variant optical elements. Because the fast axis at polar angle ξ is tilted from the x axis by ξ/2, the av-xy-Jones matrix of this av-HWP is represented as
MavHWP(ξ)=(cosξsinξsinξcosξ).
(7)

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.

When a horizontally polarized laser beam, J horizontal, passes through this angularly variant half-wave plate, the output is
MavHWP(ξ)Jhorizontal=(cosξsinξsinξcosξ)(10)=(cosξsinξ).
(8)
This is the radial polarization J radial(ξ) shown in Eq. (1).

The formula of rotation for a conventional Jones matrix is given as
M(θ)=R(θ)MR(θ).
(9)
This formula does not hold for an av-xy-Jones matrix. Correctly, under the same logic as with that of the Jones vector, the ξ in M must be replaced by ξθ. As a result, the correct formula of rotation for an av-xy-Jones matrix is represented as
M(θ)(ξ)=R(θ)M(ξθ)R(θ).
(10)
By applying this formula to the av-HWP represented by Eq. (7), we obtain the notation of the av-HWP rotated by π/2 as,
MavHWP(π/2)(ξ)=R(π/2)MavHWP(ξ)R(π/2)=(sinξcosξcosξsinξ).
(11)
When a horizontally polarized beam J horizontal passes through this, the output,
MavHWP(π/2)(ξ)Jhorizontal=(sinξcosξ),
(12)
is the azimuthal polarization J azimuthal(ξ) shown in Eq. (2). It has already been reported that both radial and azimuthal polarizations can be generated from linear polarization by using av-HWP [8

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

, 15

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]

]. This was reproduced with the proposed calculus, validating the derived rotation formula Eq. (10).

Another example of an optical element is an angularly variant linear polarizer (av-LP) with a transmission axis radial, shown in Fig. 3(b). The av-xy-Jones matrix of this element is represented as
MavLP(ξ)=(cos2ξsinξcosξsinξcosξsin2ξ),
(13)
because the transmission axis at polar angle ξ is tilted by ξ from the x axis. Rotation of this element by any angle θ must keep the matrix unchanged, because of its rotational symmetry. Namely,
MavLP(θ)(ξ)=R(θ)MavLP(ξθ)R(θ),
(14)
must be independent of θ and equal to M av–LP(ξ). We can confirm this by a long calculation.

3. Polar-Jones calculus

3.1. General

3.2. Polar-Jones vectors

The polar-Jones vector resolves the complex amplitude of the electric field into radial and azimuthal components; that is,
J̌=(Erad(ξ)Eaz(ξ)).
Here we define outward and counterclockwise directions that are positive for radial and azimuthal components, respectively. Let us consider examples. The polar-Jones vector for radial polarization shown in Fig. 1(a) is
J̌radial=(10),
(16)
and the azimuthal polarization shown in Fig. 1(b) is
J̌azimuthal=(01).
(17)

As can be seen, this notation makes it easier to recognize the characteristics of axially symmetric beams.

Conversion between xy- and polar-Jones vectors is obtained by considering the geometry as
J̌(ξ)=R(ξ)J(ξ),
(18)
J(ξ)=R(ξ)J̌(ξ).
(19)
These equations indicate that both sides are mathematically equal when (ξ) and J(ξ) describe physically the same states. It is natural that these equations are similar to the formula of rotation of conventional Jones vectors, Eq. (3), because the radial axis at polar angle ξ is tilted by angle ξ from the x axis of the Cartesian coordinate.

If necessary, a conventional (that is, not angularly variant) beam can also be represented by the polar-Jones notation. For example, a linearly polarized beam in the horizontal (x) direction, Jhorizontal=(10), is represented by the polar-Jones notation as
J̌horizontal=R(ξ)Jhorizontal=(cosξsinξ).
(20)

3.3. Polar-Jones matrices

The av-HWP with a fast axis tilted by ξ/2 from the x axis at angle ξ, shown in Fig. 3(a), is represented as
M̌avHWP=(cosξsinξsinξcosξ),
(21)
because the fast axis is tilted by −ξ/2 from the radial axis at any polar angle ξ. When a horizontally polarized laser beam, horizontal, passes through this av-HWP, the output
M̌avHWPJ̌horizontal=(cosξsinξsinξcosξ)(cosξsinξ)=(10),
(22)
is the radial polarization, radial, as shown in Eq. (16). As can be seen, the polar-Jones calculus gave the same result as that obtained by the xy-Jones calculus (Eq. (6)).

Conversion between xy- and polar-Jones matrices is derived as follows. If the conversion formula Eq. (19) is applied to M̌ J̌,
MJ=R(ξ)M̌J̌,
(23)
is obtained. Substituting Eq. (18) into this equation gives
MJ=R(ξ)M̌R(ξ)J,
(24)
where the right most operand is J on both the left and right sides. Thus, finally, the conversion formulas
M=R(ξ)M̌R(ξ),
(25)
M̌=R(ξ)MR(ξ),
(26)
are derived. Again these equations are similar to the formula of rotation of conventional Jones matrices.

Another example of an optical element is the av-LP with a transmission axis radial, shown in Fig. 3(b). The polar-Jones matrix of this element is given as a simple form
M̌avLP=(1000).
(27)
When a left-handed circularly polarized beam,
J̌LCP(ξ)=R(ξ)JLCP,
(28)
where JLCP=12(1i) is a left-handed circularly polarized beam in xy-Jones notation, passes through this av-LP, the output is obtained as
M̌avLP(ξ)J̌LCP(ξ)=M̌avLP(ξ)R(ξ)JLCP=12(cosξ+isinξ0).
(29)
This equation indicates that the direction of the electric field is everywhere radial but that the phase of oscillation varies with angle ξ. For the sake of comparison, the same polarization state is calculated as follows in xy-Jones notation. The xy-Jones matrix of this av-LP is represented by Eq. (13). When the left-handed circularly polarized beam, J LCP, passes through this av-LP, the output is
MavLPJLCP=12(cos2ξ+isinξcosξsinξcosξ+isin2ξ).
(30)
Although Eq. (29) and Eq. (30) both represent the same polarization state, it is much easier to recognize that state by using the polar notation Eq. (29) than by using Eq. (30). This is an advantage of the polar-Jones calculus.

3.4. Rotation formula

By considering the geometry, we obtain the formulas of rotation for polar-Jones vectors and matrices as
J̌(θ)(ξ)=J̌(ξθ),
(31)
M̌(θ)(ξ)=M̌(ξθ).
(32)
It is clear from these formulas that polar-Jones vectors and matrices in which the angle ξ does not explicitly appear, such as radial and azimuthal polarization (Eqs. (16) and (17)) and the av-LP (Eq. (27)), are not affected by the rotation of any angle.

3.5. Longitudinal component in focus

Using the polar-vector representation, we can estimate whether or not there is a longitudinal electric (or magnetic) field when the beam is focused. This is because the longitudinal component in focus comes from the in-phase radial component. Correspondingly, for an electric field of J̌=(Erad(ξ)Eaz(ξ)),
LE(J̌)=02πErad(ξ)dξ,
(33)
is a measure of the longitudinal electric field in focus. For example, L E for the radial polarization radial is
LE(J̌radial)=02π1dξ=2π,
(34)
which is obviously a non-zero value and the maximum value for a beam with normalized electric field strength. In contrast, even though the direction of the electric field is everywhere radial, the longitudinal component is zero for Eq. (29):
LE(M̌avLP(ξ)J̌LCP(ξ))=02π12(cosξ+isinξ)dξ=0.
(35)
The difference between Eq. (34) and Eq. (35) is ascribed to the different polar angle dependence of oscillation phase.

4. Table of xy- and Polar-Jones Representations

Table 1 summarizes xy- and polar-Jones vectors for some polarization states, and Table 2 summarizes xy- and polar-Jones matrices for some optical elements.

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

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

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5. Conclusion

We have deduced rotation formulas for the angularly variant term-added Jones calculus. These formulas allow us to routinely calculate the polarization state of an axially symmetric polarized beam, and hence to take full advantage of the Jones calculus. In addition, we have proposed an extension of the Jones calculus, which we call the polar-Jones calculus. This is an intuitive method to represent axial symmetry, and thus makes it easy to recognize the character of an axially symmetric beam. Formulas for conversion between different notations are given. In addition, we have proposed a simple method to estimate the longitudinal component when the beam is focused. These techniques will be helpful in analyzing an axially symmetric polarized beam.

Acknowledgments

The author would like to thank Prof. Y. Mizutani for the fruitful discussions and critical reading of the manuscript. This work was supported in part by KAKENHI ( 20360115).

References and links

1.

R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–493 (1941). [CrossRef]

2.

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

3.

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

4.

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

5.

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

6.

Focus Issue: Unconventional Polarization States of Light, Opt. Express 18(10), 10775–10923 (2010). [PubMed]

7.

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

8.

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

9.

T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33, 122–124 (2008). [CrossRef] [PubMed]

10.

Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18, 10828–10833 (2010). [CrossRef] [PubMed]

11.

M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 1948–1950 (1996). [CrossRef] [PubMed]

12.

K. J. Moh, X.-C. Yuan, J. Bu, R. E. Burge, and B. Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46, 7544–7551 (2007). [CrossRef] [PubMed]

13.

I. Moreno, J. A. Davis, I. Ruiz, and D. M. Cottrell, “Decomposition of radially and azimuthally polarized beams using a circular-polarization and vortex-sensing diffraction grating,” Opt. Express 18, 7173–7183 (2010). [CrossRef] [PubMed]

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]

OCIS Codes
(260.5430) Physical optics : Polarization
(260.6042) Physical optics : Singular optics

ToC Category:
Physical Optics

History
Original Manuscript: April 25, 2011
Revised Manuscript: June 5, 2011
Manuscript Accepted: June 6, 2011
Published: June 17, 2011

Citation
Shigeki Matsuo, "Matrix calculus for axially symmetric polarized beam," Opt. Express 19, 12815-12824 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-13-12815


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References

  1. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–493 (1941). [CrossRef]
  2. W.A. Shurcliff, Polarized Light: Production and Use (Harvard University Press, 1962).
  3. R.M.A. Azzam and N.M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  4. E. Hecht, “A mathematical description of polarization,” in Optics , 4th ed. (Addison Wesley, 2002), chap. 8.13, pp. 373–379.
  5. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009). [CrossRef]
  6. Focus Issue: Unconventional Polarization States of Light, Opt. Express 18(10), 10775–10923 (2010). [PubMed]
  7. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]
  8. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]
  9. T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33, 122–124 (2008). [CrossRef] [PubMed]
  10. Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18, 10828–10833 (2010). [CrossRef] [PubMed]
  11. M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. 21, 1948–1950 (1996). [CrossRef] [PubMed]
  12. K. J. Moh, X.-C. Yuan, J. Bu, R. E. Burge, and B. Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46, 7544–7551 (2007). [CrossRef] [PubMed]
  13. I. Moreno, J. A. Davis, I. Ruiz, and D. M. Cottrell, “Decomposition of radially and azimuthally polarized beams using a circular-polarization and vortex-sensing diffraction grating,” Opt. Express 18, 7173–7183 (2010). [CrossRef] [PubMed]
  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]

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