## Matrix calculus for axially symmetric polarized beam |

Optics Express, Vol. 19, Issue 13, pp. 12815-12824 (2011)

http://dx.doi.org/10.1364/OE.19.012815

Acrobat PDF (714 KB)

### 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

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

*z*direction is resolved into

*x*and

*y*components, and is represented as a column vector (the symbol

**is frequently used). As well, an optical element is represented as a 2×2 matrix (the symbol**

*J***is frequently used). The Jones calculus is a well known method for analyzing a completely polarized beam whose cross section is homogeneously polarized.**

*M*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]

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]

*kz*–

*ωt*, not

*ωt*–

*kz*.

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

### 2.1. Rotation formula for av-xy-Jones vectors

*J*_{radial}(

*ξ*) represented by Eq. (1), 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]. 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

**must be replaced by**

*J**ξ*–

*θ*. As a result, is the correct formula of rotation for an av-xy-Jones vector. If we apply this formula to the radial polarization, 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.

### 2.2. Rotation formula for av-xy-Jones matrices

*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. 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]

*ξ*is tilted from the

*x*axis by

*ξ*/2, the av-xy-Jones matrix of this av-HWP is represented as

*J*_{horizontal}, passes through this angularly variant half-wave plate, the output is This is the radial polarization

*J*_{radial}(

*ξ*) shown in Eq. (1).

*ξ*in

**must be replaced by**

*M**ξ*–

*θ*. As a result, the correct formula of rotation for an av-xy-Jones matrix is represented as 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, When a horizontally polarized beam

*J*_{horizontal}passes through this, the output, 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. 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]

*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, 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

*x*and

*y*basis). Generally, the other basis can be chosen in the Jones Calculus [3]. For an angularly symmetric system, another presumable choice of basis is the radial and azimuthal basis in the polar coordinate system. As mentioned in Sec. 1, the Jones calculus based on polar coordinates is referred to as the polar-Jones calculus. Because the polar-Jones calculus is not compatible with the xy-Jones calculus, the polar-Jones vectors and matrices are shown with a caron, such as in

**and**

*J̌***. In this section, we discuss the notation, rotation formula, and conversion formula between the xy-Jones calculus and the polar-Jones calculus.**

*M̌*### 3.2. Polar-Jones vectors

**(**

*J̌**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.

*x*) direction,

### 3.3. Polar-Jones matrices

*ξ*/2 from the

*x*axis at angle

*ξ*, shown in Fig. 3(a), is represented as because the fast axis is tilted by −

*ξ*/2 from the radial axis at any polar angle

*ξ*. When a horizontally polarized laser beam,

*J̌*_{horizontal}, passes through this av-HWP, the output is the radial polarization,

*J̌*_{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)).

**, is obtained. Substituting Eq. (18) into this equation gives where the right most operand is**

*M̌ J̌***on both the left and right sides. Thus, finally, the conversion formulas are derived. Again these equations are similar to the formula of rotation of conventional Jones matrices.**

*J**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 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

*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

*L*

_{E}for the radial polarization

*J̌*_{radial}is 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): 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

## 5. Conclusion

## Acknowledgments

## References and links

1. | R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. |

2. | W.A. Shurcliff, |

3. | R.M.A. Azzam and N.M. Bashara, |

4. | E. Hecht, “A mathematical description of polarization,” in |

5. | Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. |

6. | Focus Issue: Unconventional Polarization States of Light, Opt. Express |

7. | S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. |

8. | R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. |

9. | T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. |

10. | Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express |

11. | M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett. |

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. |

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 |

14. | Noted that |

15. | G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett. |

**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

- R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–493 (1941). [CrossRef]
- 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.
- Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009). [CrossRef]
- Focus Issue: Unconventional Polarization States of Light, Opt. Express 18(10), 10775–10923 (2010). [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]
- R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]
- 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]
- Y. Kozawa and S. Sato, “Optical trapping of micrometer-sized dielectric particles by cylindrical vector beams,” Opt. Express 18, 10828–10833 (2010). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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.
- 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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