## Manipulation of the Pancharatnam phase in vectorial vortices

Optics Express, Vol. 14, Issue 10, pp. 4208-4220 (2006)

http://dx.doi.org/10.1364/OE.14.004208

Acrobat PDF (3723 KB)

### Abstract

Linearly polarized vectorial vortices are analyzed according to their Pancharatnam phase and experimentally demonstrated using a geometric phase element consisting of space-variant subwavelength gratings. It is shown that in the absence of a Pancharatnam phase, stable vectorial vortices that have no angular momentum arise. In contrast, if a Pancharatnam phase is present the vectorial vortices have orbital angular momentum and collapse upon propagation.

© 2006 Optical Society of America

## 1. Introduction

2. D. Palacios, D. Rozas, and G. A. Swartzlander Jr., “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. **88**, 103902 1–4 (2002). [CrossRef]

3. J.F. Nye, “Polarization effect in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. Lond. A **387**, 105–132 (1983). [CrossRef]

7. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. **213**, 201–221 (2002). [CrossRef]

8. P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, and F. Gori, “General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields,” Opt. Express **10**, 949–959 (2002). [PubMed]

9. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Propagation-invariant vectorial Bessel beams obtained by use of quantized Pancharatnam-Berry phase optical elements,” Opt. Lett. **29**, 238–240 (2004). [CrossRef] [PubMed]

10. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Rotating vectorial vortices produced by space-variant subwavelength gratings,” Opt. Lett. **30**, 2933–2935, (2005). [CrossRef] [PubMed]

11. D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astro. Phys. **633**, 1191–1200 (2005). [CrossRef]

12. Y. Liu, D. Cline, and P. He, “Vacuum laser acceleration using a radially polarized CO_{2} laser beam,” Nucl. Instrum. Meth. Phys Res. A424, 296–303 (1999);
W. D. Kimura, G. H. Kim, R. D. Romea, L. C. Steinhauer, I. V. Pogorelsky, K. P. Kusche, R. C. Fernow, X. Wang, and Y. Liu, “Laser acceleration of relativistic electons using the inverse Cherenkov effect,” Phys. Rev. Lett.74, 546–549 (1995). [CrossRef]

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

14. 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]

15. R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. **77**, 3322–3324 (2000). [CrossRef]

16. K. C. Toussaint Jr., S. Park, J. E. Jureller, and N. F. Scherer, “Generation of optical vector beams with a diffractive optical element interferometer,” Opt. Lett. **30**, 2846–2848 (2005). [CrossRef] [PubMed]

9. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Propagation-invariant vectorial Bessel beams obtained by use of quantized Pancharatnam-Berry phase optical elements,” Opt. Lett. **29**, 238–240 (2004). [CrossRef] [PubMed]

10. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Rotating vectorial vortices produced by space-variant subwavelength gratings,” Opt. Lett. **30**, 2933–2935, (2005). [CrossRef] [PubMed]

17. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Formation of linearly polarized light with axial symmetry by use of space-variant subwavelength gratings,” Opt. Lett. **28**, 510–512 (2003). [CrossRef] [PubMed]

18. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. **27**, 1141–1143 (2002). [CrossRef]

19. Q. Zhan and J. R. Leger, “Interferometric measurement of the geometric phase in space-variant polarization manipulations,” Opt. Commun. **213**, 241–245 (2002). [CrossRef]

_{2}laser radiation of 10.6

*µm*wavelength. Chapter 4 describes the experimental demonstration verifying the theoretical analysis of the vectorial vortices generated by these space-variant subwavelength gratings. This verification was achieved by measuring the full polarization state at the immediate outlet of the devices, and at their Fraunhofer diffraction. Our concluding remarks are presented in Chapter 5.

## 2. Theory

### 2.1 The structure of vectorial vortices and their Pancharatnam phase

*m*and

*n*are integers, |

*R*〉 and |

*L*〉 are the right- and left-handed polarization unit vectors, respectively, and

*φ*is the angle in the polar coordinate system (

*r,φ*). Equation (1) is consistent with the definition of a vectorial vortex as defined above [10

10. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Rotating vectorial vortices produced by space-variant subwavelength gratings,” Opt. Lett. **30**, 2933–2935, (2005). [CrossRef] [PubMed]

*φ*

_{0}represents arbitrary retardation. As linearly polarized vortices are the sole topic of this paper, the term “vectorial vortex” will henceforth serve as the abbreviation for “linearly polarized vectorial vortices”. Figure 1 illustrates the polarization state of vectorial vortices with different values of

*m*and

*n*, with

*φ*

_{0}=0; the polarization ellipses have degenerated and thus appear as bars. Usually, vectorial singularities are studied by looking at the temporal evolution of the field [3

3. J.F. Nye, “Polarization effect in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. Lond. A **387**, 105–132 (1983). [CrossRef]

4. J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London Ser. A **389**, 279–290 (1983). [CrossRef]

17. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Formation of linearly polarized light with axial symmetry by use of space-variant subwavelength gratings,” Opt. Lett. **28**, 510–512 (2003). [CrossRef] [PubMed]

*A*〉 and |

*B*〉 is defined according to

*φ*=arg〈

_{P}*A*|

*B*〉. Note that a

*π*phase of indeterminate sign appears in

*φ*

_{P}across point where 〈

*A*|

*B*〉=0. However, as these steps are of no special physical significance, they will be omitted from the following discussion altogether. Taking |

*E*(

*φ*=0)〉 as a reference, the Pancharatnam phase of the vectorial vortex is given by

*m-n*)

*φ*/2, [as calculated from Eq. (1)], but are also advanced or retarded according to Eq. (2).

*C*, encircles the phase singularity. We denote the topological charge of the vectorial vortex as a topological Pancharatnam charge. Applying Eq. (2) to Eq. (4), the topological Pancharatnam charge of the vectorial vortex is

*l*=(

_{P}*m+n*)/2. To establish the connection between

*l*and the angular momentum of the vectorial vortex, let us first calculate the total angular momentum of the vectorial vortex as the sum of the orbital angular momentum of its scalar components. The spin angular momentum is canceled out as indicated by Eq. (1). This calculation results in a normalized angular momentum of,

_{P}*P*is the total intensity of the field and

*ω*is the optical frequency. Comparing this result to the expression for the topological Pancharatnam charge of a vectorial vortex, we find,

*z*is the propagation distance,

*k*is the wavenumber, and

*R*corresponds to the radius of the finite aperture of the field. Equation (7) shows that the components of the vectorial vortex undergo different modulations as the distance

*z*is increased. As a result, a vectorial vortex does not maintain its structure upon propagation. However, in the unique case where no Pancharatnam phase is present, i.e.,

*n*=-

*m*[see Eq. (2)], Eq. (7) reduces to,

### 2.2 Geometric phase elements

24. L. H. Cescato, E. Gluch, and N. Streibl, “Holographic quarterwave plates,” Appl. Opt. **29**3286–3290 (1990). [CrossRef] [PubMed]

25. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Spiral phase elements obtained by use of discrete space-variant subwavelength gratings,” Opt. Commun. **251**, 306–314 (2005). [CrossRef]

*E*〉 represents the beam impinging on the device and

_{in}*θ*=

*θ*(

*x,y*) is the local orientation of the subwavelength grooves.

*tx*,

*ty*are the amplitude transmission coefficients for light polarized perpendicular and parallel to the subwavelength grooves, respectively, and

*ϕ*is the retardation phase. Equation (9) indicates that the field emerging from a space-variant subwavelength grating comprises three components. The first maintains the original polarization state and phase of the incoming beam. The second is right-handed circularly polarized and has a phase modification of 2

*θ*(

*x,y*). The third has an orthogonal polarization direction and opposite phase modification with respect to the second component. Note that the magnitude of the different components is determined by the local birefringent parameters

*t*and

_{x}, t_{y}*ϕ*, as well as by the incoming polarization state for the second and third components. The transmission of dielectric gratings is relatively high and the retardation

*ϕ*is primarily a function of the subwavelength grooves etching depth. Therefore, we consider devices with subwavelength grooves for which

*t*≈

_{x}*t*≈1 and

_{y}*ϕ*=

*π*or

*π*/2, i.e. perfect space-variant half and quarter wave plates.

*θ*

_{0}=

*ϕ*/2,

*t*=1, and

_{x}=t_{y}*ϕ=π*, for linearly polarized illumination is,

*n*=-

*m*. Thus, a vectorial vortex with field vectors which has no Pancharatnam phase is produced. From the discussion in chapter 2.1, this vectorial vortex exhibits no orbital angular momentum and has beam-like propagation.

*t*=1 and

_{x}=t_{y}*ϕ=π*/2, with subwavelength groove orientation given by Eq. (10). In this case, if

*θ*

_{0}=

*π*/4 and the illuminating beam is left- handed circularly polarized, then according to Eq. (9) the emerging field is

*n*=0. This is a vectorial vortex with Pancharatnam phase of helical structure, thus

*φ*=

_{P}*mφ*/2 as can be calculated from Eq. (2). From Eq. (5), this vectorial vortex has an orbital angular momentum of

*ħm*/2 per photon. However, from Eq. (7), we find that it does not maintain its polarization structure upon propagation. Note, in case of m=±1 the central singularity is a generic feature of vectorial fields known as C-point [4

4. J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London Ser. A **389**, 279–290 (1983). [CrossRef]

*θ*

_{0}is of no special importance apart from the case of the space-variant quarter wave plate with

*m*=2.

_{1}, ŝ

_{2}, ŝ

_{3}serve as rectangular coordinates [26]. In this representation, a specific polarization state is mapped to a point on the sphere, while polarization state transformations are represented by geodesic lines connecting the initial and final polarization states. Let us consider the geodesic triangle ABC of Fig. 2(a) using similar calculations to those performed by Aravind [24

24. L. H. Cescato, E. Gluch, and N. Streibl, “Holographic quarterwave plates,” Appl. Opt. **29**3286–3290 (1990). [CrossRef] [PubMed]

27. P. K. Aravind, “A simple proof of Pancharatnam’s theorem,” Opt. Commun.94, 191–196 (1992);
C. Brosseau, *Fundamentals of Polarized Light* (Wiley, New York, 1998). [CrossRef]

*n*=0 and

*φ*

_{0}=0) that in this case as well, A is in phase with B and with C. According to the geometric considerations given in Ref. 27, we find the area of the geodesic triangle ABC (shaded in the figure) to be

*mφ*stradians. Therefore, a geometric phase that equals half the area enclosed on the Poincaré sphere by the geodesic lines is added to the wave at C with respect to B. Comparing these results with Eq. (2), we conclude that vectorial vortices that are generated by space-variant subwavelength gratings have an entirely geometric Pancharatnam phase.

## 3. Design and realization of subwavelength gratings for the generation of vectorial vortices

17. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Formation of linearly polarized light with axial symmetry by use of space-variant subwavelength gratings,” Opt. Lett. **28**, 510–512 (2003). [CrossRef] [PubMed]

25. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Spiral phase elements obtained by use of discrete space-variant subwavelength gratings,” Opt. Commun. **251**, 306–314 (2005). [CrossRef]

*N*=16 [25

25. A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Spiral phase elements obtained by use of discrete space-variant subwavelength gratings,” Opt. Commun. **251**, 306–314 (2005). [CrossRef]

*θ*

_{0}=

*π*/4 and

*m*=1, 2, 3, 4 were fabricated. The masks were 10

*mm*in diameter and had

*N*=16 zones. A subwavelength period of Λ=2

*µm*was chosen along with a fill factor of 0.5 for use with 10.6

*µm*wavelength radiation. The masks were transferred by contact lithography to 500

*µm*thick GaAs wafers and space-variant subwavelength gratings were achieved using the fabrication process described in Ref. 25. The nominal etching depths were 2.5

*µm*and 5

*µm*, in order to achieve the desired

*π*/2 and

*π*-retardation, respectively. As a final step, the backsides of the elements were applied with an anti-reflection coating. Figure 3 shows scanning electron microscope images of the devices. Discrete changes in the groove orientation as well as the high aspect ratio and rectangular shape of the grooves are clearly observed. For a device with nominal retardation of

*π*radians, we have previously measured the amplitude transmission to be

*t*=0.74 and

_{x}*t*=0.86, with actual retardation of

_{y}*ϕ*=0.97

*π*[25

**251**, 306–314 (2005). [CrossRef]

## 4. Experimental results

*π*-retardation devices with 10.6

*µm*linearly polarized light from a CO

_{2}laser. Figure 4(a) shows the intensity distributions at the immediate outlet of the devices when imaged through a linear polarizer. The fringes indicate the rotation of the polarization ellipses according to Eq. (3) for

*n*=-

*m*. We have measured the polarization state distribution of the vectorial vortices using the four-measurement technique [28]. Figure 4(b) shows the azimuthal angle distribution. The rotation around the field axis is clearly observed. We have found the typical deviation of the azimuthal angle with respect to its desired value to be less than 2% (0.026 radians). The typical ellipticity of the emerging field was less than 0.07 radian. This result is comparable with the expected performance of a device with

*N*=16, indicating the excellent ability of a space-variant subwavelength grating to control the polarization state of a beam. Vectorial vortices having a Pancharatnam phase were obtained by illuminating the

*π*/2-retardation devices with 10.6

*µm*left-handed circularly polarized light from a CO

_{2}laser. Figure 4(c) shows the intensity distributions at the immediate outlet of the devices imaged through a linear polarizer. In this case, the fringes indicate rotation of the polarization ellipses that is in agreement with Eq. (3) for

*n*=0. Figure 4(d) shows the measured azimuthal angle in this case. Typical values for the deviation of the azimuthal angle and ellipticity, compared to their desired value, are similar to the former case, thus indicating the formation of high quality vectorial vortices.

### 4.1 Fraunhofer diffraction of vectorial vortices without Pancharatnam phase

*n*=-

*m*) were obtained at the focus of a lens with 1

*m*focal length. Figure 5(a) shows their measured intensity distributions. The annular intensity pattern that is predicted by Eq. (8) is clearly observed. Another way to understand this is to average the fields located on a circle surrounding the singularity and limiting the circle radius to 0, thus

*E*(

*r,φ*), is calculated from Eq. (8). This average, in the presented case, results in

*E*̃=(0,0)

^{T}, where

*T*denotes transposition. Thus, the dark core is a result of destructive interference of the field at the center. This outcome is also predicted by Eq. (1), when considering conjugate scalar vortices embedded in both orthogonally polarized components of the vectorial vortex. Typical cross sections of the intensity distributions are given in Fig. 5(b). Good agreement between the experimental results and the theoretical analysis is obtained. Moreover, Fig. 5(c) shows the measured space-variant azimuthal angle of the beam’s polarization state at the Fraunhofer region. The close resemblance of the polarization states between the near- and the far-fields validates our conclusion that vectorial vortices that do not have a Pancharatnam phase maintain their structure upon propagation.

### 4.2 Fraunhofer diffraction of vectorial vortices with Pancharatnam phase

*m*focal length, and are shown in Fig. 6(a). In this case,

*n*=0 and a bright spot at the center of the field is observed. The bright central spots are also shown in the typical cross sections of Fig. 6(b). These bright central spots (as well as the annular intensity rings) are anticipated from Eq. (7) for

*n*=0. This results from the constructive interference at the center of the field. The central spots’ polarization state can be found by applying Eq. (7) to Eq. (15), yielding

*E*̃∝|

*L*〉. The measured polarization ellipses of the far-field vectorial vortices are shown in Fig. 6(c). Different colors indicate different handedness of the field. At the boundary between handedness, there is a line of linear polarization known as an L-line [6

6. I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. **201**, 251–270 (2002). [CrossRef]

7. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. **213**, 201–221 (2002). [CrossRef]

### 4.3 Fraunhofer diffraction of general vectorial vortex

*m,n*≠0as well as

*m*≠-

*n*, (see Eq. (1) that has a Pancharatnam phase, by combining a spiral phase element with a scalar topological charge,

*l*, immediately behind the

*π*-retardation device. In this case, a phase of

*lφ*is added to both components of the beam producing

*l*=2 and a

*π*-retardation device of

*m*=3. The spiral phase element was formed by a 32-level reactive-ion etching on a ZnSe substrate. Figures 7(a) and 7(b) show the Fraunhofer diffraction intensity distribution of the measured vectorial vortex, as well as its measured and predicted cross sections. As can be seen from these figures, the dark central spot is obvious, as anticipated by Eqs. (7) and (15). The measured polarization ellipses of this field are shown in Fig. 7(c). As in Fig. 6, the L-line at the boundary between different handedness is clearly shown. We have measured the polarization states of both intensity rings to be left- and right-handed circularly polarized, for the inner and outer rings, respectively. This result agrees with Eq. (7). As can be seen, the measured polarization state is no longer a linearly polarized axially symmetric vectorial vortex. Hence, a vectorial vortex with a Pancharatnam phase collapses upon propagation, as discussed in chapter 2.

## 5. Conclusions

## Appendix A – The temporal evolution of the vectorial vortex

*m*+

*n*lines of zero magnitude rotating at 2

*ω*/(

*m*+

*n*) radian per second. These zero lines are known as disclinations [4

4. J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London Ser. A **389**, 279–290 (1983). [CrossRef]

*m*and

*n*. Note in Figs. 1A (a)–(c) that the orientation of the field vector and the lines of zero magnitude obey Eq. (1A). However, special attention should be given to particular cases. First, if

*m*=-

*n*, no rotating zero lines appear, but rather the field vectors vanish simultaneously, as can be seen in Fig. 1A(d). The second case is m=n, which corresponds to a uniformly oriented linearly polarized field. This case was vastly treated within the framework of scalar singular optics, thus it is omitted from the current discussion. The temporal evolution is in agreement with the concept of the Pancharatnam phase of the field. Equation (2) shows that the Pancharatnam phase advances the wave in a helical manner around the field axis, causing the location of instantaneous zeros to rotate in time, as shown in Figs. 1A(a–c). In the particular case where

*m*=-

*n*, the Pancharatnam phase is zero, and thus the beating of the waves are synchronized in time and the field vanishes simultaneously, as can be seen in Fig. 1A(d).

## References and links

1. | M. S. Soskin and M.V. VasnetsovE. Wolf, “Singular optics,” in |

2. | D. Palacios, D. Rozas, and G. A. Swartzlander Jr., “Observed scattering into a dark optical vortex core,” Phys. Rev. Lett. |

3. | J.F. Nye, “Polarization effect in the diffraction of electromagnetic waves: the role of disclinations,” Proc. R. Soc. Lond. A |

4. | J. F. Nye, “Lines of circular polarization in electromagnetic wave fields,” Proc. R. Soc. London Ser. A |

5. | J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves,” Proc. R. Soc. Lond. A414, 433–446 and 447–468 (1987). [CrossRef] |

6. | I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. |

7. | M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statistics,” Opt. Commun. |

8. | P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, and F. Gori, “General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields,” Opt. Express |

9. | A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Propagation-invariant vectorial Bessel beams obtained by use of quantized Pancharatnam-Berry phase optical elements,” Opt. Lett. |

10. | A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Rotating vectorial vortices produced by space-variant subwavelength gratings,” Opt. Lett. |

11. | D. Mawet, P. Riaud, O. Absil, and J. Surdej, “Annular groove phase mask coronagraph,” Astro. Phys. |

12. | Y. Liu, D. Cline, and P. He, “Vacuum laser acceleration using a radially polarized CO |

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

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

15. | R. Oron, S. Blit, N. Davidson, A. A. Friesem, Z. Bomzon, and E. Hasman, “The formation of laser beams with pure azimuthal or radial polarization,” Appl. Phys. Lett. |

16. | K. C. Toussaint Jr., S. Park, J. E. Jureller, and N. F. Scherer, “Generation of optical vector beams with a diffractive optical element interferometer,” Opt. Lett. |

17. | A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Formation of linearly polarized light with axial symmetry by use of space-variant subwavelength gratings,” Opt. Lett. |

18. | Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Space-variant Pancharatnam-Berry phase optical elements with computer-generated subwavelength gratings,” Opt. Lett. |

19. | Q. Zhan and J. R. Leger, “Interferometric measurement of the geometric phase in space-variant polarization manipulations,” Opt. Commun. |

20. | E. Hasman, G. Biener, A. Niv, and V. KleinerE. Wolf, “Space-variant polarization manipulation,” in |

21. | L. Allen, M.J. Padgett, and M. BabikerE. Wolf in |

22. | M. Born and E. Wolf |

23. | R. C. Enger and S.K. Case, “Optical elements with ultrahigh spatial-frequency surface corrugations,” Appl. Opt. |

24. | L. H. Cescato, E. Gluch, and N. Streibl, “Holographic quarterwave plates,” Appl. Opt. |

25. | A. Niv, G. Biener, V. Kleiner, and E. Hasman, “Spiral phase elements obtained by use of discrete space-variant subwavelength gratings,” Opt. Commun. |

26. | S. Pancharatnam, “Generalized theory of interference and its applications. Part I. Coherent pencils,” Proc. Ind. Acad. Sci. A44 (1956) 247 [reprinted in
S. Pancharatnam, |

27. | P. K. Aravind, “A simple proof of Pancharatnam’s theorem,” Opt. Commun.94, 191–196 (1992);
C. Brosseau, |

28. | E. Collett, |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(260.5430) Physical optics : Polarization

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: April 3, 2006

Revised Manuscript: May 2, 2006

Manuscript Accepted: May 2, 2006

Published: May 15, 2006

**Citation**

Avi Niv, Gabriel Biener, Vladimir Kleiner, and Erez Hasman, "Manipulation of the Pancharatnam phase in vectorial vortices," Opt. Express **14**, 4208-4220 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-10-4208

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### References

- M. S. Soskin, M.V. Vasnetsov, "Singular optics," in Progress in Optics, Vol.42, E. Wolf ed. (Elsevier, Netherlands, Amsterdam, 2001), pp. 219-276.
- D. Palacios, D. Rozas, and G. A. SwartzlanderJr., "Observed scattering into a dark optical vortex core," Phys. Rev. Lett. 88, 103902 1-4 (2002). [CrossRef]
- J.F. Nye, "Polarization effect in the diffraction of electromagnetic waves: the role of disclinations," Proc. R. Soc. Lond. A 387, 105-132 (1983). [CrossRef]
- J. F. Nye, "Lines of circular polarization in electromagnetic wave fields," Proc. R. Soc. London Ser. A 389, 279-290 (1983). [CrossRef]
- J. V. Hajnal, "Singularities in the transverse fields of electromagnetic waves," Proc. R. Soc. Lond. A 414, 433-446 and 447-468 (1987). [CrossRef]
- I. Freund, "Polarization singularity indices in Gaussian laser beams," Opt. Commun. 201, 251-270 (2002). [CrossRef]
- M. R. Dennis, "Polarization singularities in paraxial vector fields: morphology and statistics," Opt. Commun. 213, 201-221 (2002). [CrossRef]
- P. Pääkkönen, J. Tervo, P. Vahimaa, J. Turunen, and F. Gori, "General vectorial decomposition of electromagnetic fields with application to propagation-invariant and rotating fields," Opt. Express 10, 949-959 (2002). [PubMed]
- A. Niv, G. Biener, V. Kleiner, and E. Hasman, "Propagation-invariant vectorial Bessel beams obtained by use of quantized Pancharatnam-Berry phase optical elements," Opt. Lett. 29, 238-240 (2004). [CrossRef] [PubMed]
- A. Niv, G. Biener, V. Kleiner, and E. Hasman, "Rotating vectorial vortices produced by space-variant subwavelength gratings," Opt. Lett. 30, 2933-2935, (2005). [CrossRef] [PubMed]
- D. Mawet, P. Riaud, O. Absil, and J. Surdej, "Annular groove phase mask coronagraph," Astro. Phys. 633, 1191-1200 (2005). [CrossRef]
- 12. Y. Liu, D. Cline, and P. He, "Vacuum laser acceleration using a radially polarized CO2 laser beam," Nucl. Instrum. Meth. Phys Res. A 424, 296-303 (1999);W. D. Kimura, G. H. Kim, R. D. Romea, L. C. Steinhauer, I. V. Pogorelsky, K. P. Kusche, R. C. Fernow, X. Wang, and Y. Liu, "Laser acceleration of relativistic electons using the inverse Cherenkov effect," Phys. Rev. Lett. 74, 546-549 (1995). [CrossRef]
- S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "Focusing light to a tighter spot," Opt. Commun. 179, 1-7 (2000). [CrossRef]
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