## Generation of arbitrary spatially variant polarization beams with a trapezoid Sagnac interferometer |

Optics Express, Vol. 20, Issue 19, pp. 21715-21721 (2012)

http://dx.doi.org/10.1364/OE.20.021715

Acrobat PDF (2026 KB)

### Abstract

We present a flexible approach to generate arbitrary vector beams with a trapezoid Sagnac interferometer. With the interferometer, the different orders of two orthogonally polarized beams from computer-generated holograms coincide with each other in Fourier spectrum domain, and coaxially combine into the vector beams. This approach provides convenient way to experimentally study the properties of vector beams with complex polarization.

© 2012 OSA

## 1. Introduction

## 2. Experimental setup

^{+}laser with wavelength of

*λ*= 514.5nm is focused on a rotating diffuser (RD, rotating frosted glass plate) and then collimated, with its polarization state rotated by a half-wave (

*λ*/2) plate. The RD is used to integrate the changing speckles over time and generate the partially coherent light field with homogeneous intensity distribution. Actually, the RD can be removed from the setup if a high quality of optical field is not required. The input collimated beam is split by a polarizing beam splitter (PBS) into two orthogonally polarized beams, i.e. the vertically (

*s*-component) and horizontally (

*p*-component) polarized ones. The ratio of the two components is adjusted to be 1:1 by rotating the half-wave plate. In Fig. 1, the PBS and three mirrors (M

_{2}, M

_{3}and M

_{4}) compose a Sagnac interferometer, and the

*s*- and

*p*-components are combined by the PBS after passing through the same path

*a*-

*b*-

*c*-

*d*-

*a*. The coaxiality of the two components is easily achieved by adjusting the angle of an arbitrary mirror. Once the coaxiality is achieved, it will be insensitive to the initial conditions of the input beam (such as the incident angle and the collimation degree). A CGH (see the inset of CGH in Fig. 1) is placed at the position

*e*, which ensures that the optical paths of the two components after passing through the CGH are equal, i.e.

*ad*+

*de*=

*ab*+

*bc*+

*ce*. This makes the path consisted of the PBS and the three mirrors has to be a trapezoid rather than a rectangle. If the position of the CGH mismatches the position

*e*, an additional phase different

*φ*

_{0}between

*s*- and

*p*-components will be attached. As a result, the CGH position can be slightly adjusted around position

*e*to control the phase different. After combined by the PBS, the ± 1 orders of the

*s*- and

*p*-components are filtered through the same open aperture F placed on the Fourier plane of L

_{3}(the aperture positions for different CGHs are marked by the dotted circles in Fig. 1), and are respectively converted into the left- and right-hand circularly polarized beams by a quarter-wave (

*λ*/4) plate (with its

*c*-axis tilted 45° relative to the horizontal). The output field is detected by a CCD camera in the image plane of the CGH. It deserves special mention that, to make sure the optical paths of

*s*- and

*p*-components after the CGH being equal, the positions of the CGH and the lens L

_{3}need to be adjusted until the CGH images on the CCD with the two components are both clear.

*s*- and

*p*-components. Although the two components could be changed back into the

*s*- and

*p*-polarizations via the PBS, the polarization conversions of the two components are not equivalent, finally leading to the distortion of the superposed polarizations. To avoid this distortion, two film polarizers are posted on both facets of the SLM, with the polarization axis at a 45° angle to the horizontal. As a result, the intensities of

*s*- and

*p*- components are both weakened 75%, and the generating efficiency of vector beam is greatly reduced. In general, a designed CGH etched on the glass or captured in the film is much more efficient.

## 3. Results and discussions

### 3.1 Generating with one-dimensional computer-generated holograms (CGHs)

22. X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. **32**(24), 3549–3551 (2007). [CrossRef] [PubMed]

*δ*is the additional phase distribution, and

*D*is the grating constant of the CGH.

*s*-component reads

*t*, while the

_{s}= t*p*-component reads

*t*[1 + cos(2π

_{p}=*y*/

*D*+

*δ*

_{−})]/2, where

*δ*

_{−}=

*δ*(−

*x*,

*y*). After passing through the trapezoid Sagnac interferometer, the ± 1 orders of the

*s*-component just overlap that of the

*p*-component, respectively (see the inset of Fourier spectrum from 1D CGH in Fig. 1), with phase difference

*φ*

_{0}. Only one order, for instance the + 1 order [of which the phase distributions for the

*s*- and

*p*-components are exp(i

*δ*) and exp(i

*δ*

_{−}+ i

*φ*

_{0}), respectively], is allowed to pass through. After converted to circular polarizations with

*λ*/4 plate, the optical fields of the two components can be expressed aswhere

*A*

_{0}is the constant factor. Then the superposed field iswhich can describe an arbitrary linear polarization distribution. By designing the phase distribution

*δ*of the CGH and adjusting the phase difference

*φ*

_{0}, arbitrary spatially variant polarization beams can be obtained. However, an unexpected phase distribution exp[i(

*δ*+

*δ*

_{−}+

*φ*

_{0})/2] is attached to the generated beam. To avoid the influence of additional phase distribution, the phase function is designed to satisfy

*δ*

_{−}= −

*δ*+

*δ*

_{0}, where

*δ*

_{0}is the additional constant phase. For example, when

*δ = mφ*+

*δ*

_{0}(where

*φ*is the azimuth angle, and

*m*is the topological charge),

*δ*

_{−}= −

*mφ*+

*m*π +

*δ*

_{0}, then a cylindrical vector beam [cos(

*mφ*-

*m*π/2-

*φ*

_{0}/2), sin (

*mφ*-

*m*π/2-

*φ*

_{0}/2)]

^{T}is formed.

*δ = φ*,

*φ*

_{0}= -π, and

*δ =*2

*φ*,

*φ*

_{0}= 0, respectively. While for the double-modes, there contain two types of polarization states separated by the circular line

*r*=

*r*

_{0}/2 (

*r*is the radial coordinate,

*r*

_{0}is the radius of the circular field). In Fig. 2(c), the parameters of CGH are

*δ =*-

*φ*and

*φ*for inside- and outside-modes, respectively, and

*φ*

_{0}= -π. In Fig. 2(d), the parameters are

*δ =*2

*φ*and

*φ*for inside and outside modes, respectively, and

*φ*

_{0}= 0. The top of Fig. 2 shows the vector beams with homogeneous intensity distributions. Due to the singularity of the polarization, the center of each field presents a dark spot. Especially in the light fields of double-modes, the regions of singular polarizations present dark rings. To analyze the polarization of the generated fields, polarization analyzers (see Fig. 1) with the axis at 0° and 45° angle to the horizontal are employed, and the corresponding intensity distributions are depicted in the middle and bottom of Fig. 2, respectively. The results are in good agreement with the theoretical polarizations. Furthermore, if we replace the

*λ*/4 plate with a

*λ*/2 plate, and set the

*c*-axis on a 22.5° angle relative to the horizontal, a special vector beam with hybrid polarization [cos(

*mφ*-

*m*π/2-

*φ*

_{0}/2),i sin (

*mφ*-

*m*π/2-

*φ*

_{0}/2)]

^{T}can be obtained [25

25. X. L. Wang, Y. Li, J. Chen, C. S. Guo, J. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express **18**(10), 10786–10795 (2010). [CrossRef] [PubMed]

### 3.2 Generating with two-dimensional CGHs

26. H. Chen, J. Hao, B. F. Zhang, J. Xu, J. Ding, and H. T. Wang, “Generation of vector beam with space-variant distribution of both polarization and phase,” Opt. Lett. **36**(16), 3179–3181 (2011). [CrossRef] [PubMed]

*δ*

_{2−}(

*x*,

*y*) =

*δ*

_{2}(−

*x*,

*y*). The 2D CGH can be considered as the superposition of two orthogonal oblique 1D CGHs [see the top of Figs. 3(a) and 3(b), with the amplitude transmission function

*t*

_{1}

*=*{1 + cos[2π(

*x*+

*y*)/

*D*+

*δ*

_{1}]}/4 and

*t*

_{2}

*=*{1 + cos[2π(

*x*-

*y*)/

*D*+

*δ*

_{2−}]}/4, respectively]. In particular, the ± 1 orders of the Fourier spectrum from the 2D CGH can be considered as the superposition of the spectra from the two 1D CGHs. The Fourier spectra of the two oblique CGHs (with ± 1 orders of the

*s*- and

*p*-components denoted by ± 1

*and ± 1*

_{p}*, respectively) are illustrated in the bottoms of Figs. 3(a) and 3(b), respectively. It reveals that the*

_{s}*p*-polarized ± 1 order from the CGH

*t*

_{1}coincides with the

*s*-polarized ± 1 order from

*t*

_{2}. Here, the influence of the phase difference

*φ*

_{0}is identical to the case with 1D CGH. To simplify the discussion, we adjust the phase difference

*φ*

_{0}as a constant, e.g.

*φ*

_{0}= 0. Filtered by an open aperture in the Fourier plan, the + 1 orders of the

*s-*and

*p-*components from the 2D CGH [with the phase distributions exp(i

*δ*

_{1}) and exp(i

*δ*

_{2}), respectively] pass through and translate into left- and right-hand circularly polarized beams, i.e. the two components can be respectively expressed as

*φ*), as shown in Fig. 3(c), is generated with

*δ*

_{1}

*=*2

*φ*+ π/2,

*δ*

_{2}

*=*−π/2 for the inside mode and

*δ*

_{1}

*=*3

*φ*,

*δ*

_{2}

*=*−

*φ*for the outside mode. Since the phase structures of these vector beams have been verified [25

25. X. L. Wang, Y. Li, J. Chen, C. S. Guo, J. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express **18**(10), 10786–10795 (2010). [CrossRef] [PubMed]

*r*, and the parameters for CGHs are

*δ*

_{1}

*= φ*+ π,

*δ*

_{2}

*=*2.5π

*r*/

*r*

_{0}(

*r*

_{0}is the radius of the circular field) and

*δ*

_{1}

*=*3π

*r*/

*r*

_{0},

*δ*

_{2}

*=*−3π

*r*/

*r*

_{0}, respectively. The polarization state in Fig. 3(d) has azimuthal and radial dependence, similar to the phase structure of the helical-conical beam, which presents peculiar focusing property [28

28. C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express **13**(5), 1749–1760 (2005). [CrossRef] [PubMed]

8. X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. **105**(25), 253602 (2010). [CrossRef] [PubMed]

*λ*/4 plate is removed, and an additional intensity modulation is attached to the 2D CGH, of which the amplitude transmission function is

*t*(

*x*,

*y*)

*=*{2 +

*A*

_{1}cos[2π(

*x*+

*y*)

*/D*+

*δ*

_{1}] +

*A*

_{2}cos[2π(

*x*−

*y*)

*/D*+

*δ*

_{2}]}/4. Then the superposed field is proportional to

*A*

_{1}exp(i

*δ*

_{1})[1,0]

^{T}+

*A*

_{2}exp(i

*δ*

_{2})[0,1]

^{T}

*=*exp(i

*δ*

_{1})[

*A*

_{1},

*A*

_{2}exp(i

*δ*

_{2}-i

*δ*

_{1})]

^{T}. It can describe an arbitrary (including linear, circular and elliptical) polarization distribution. For a radially polarized beam,

*A*

_{1}

*=*cos

*φ*,

*A*

_{2}exp(i

*δ*

_{2}-i

*δ*

_{1})

*=*sin

*φ*. However, due to the effect of aperture filter, some spectral information of the CGH lost, and the intensity modulation (

*A*

_{1}and

*A*

_{2}) cannot be exactly reconstructed via the holography. As a result, the vector beams are difficult to be perfectly created. Especially with partially coherent light, the distortion of reconstructed light field is more intense. Thus, we remove the rotating diffuser, and obtain two vector beams as shown in Fig. 4 , where 4(a) and 4(b) correspond to the radial [see Fig. 2(a)] and radial-variant [see Fig. 3(e)] polarizations, respectively. The transmission functions of the CGH used in Figs. 4(a) and 4(b) are

*t*(

*x*,

*y*)

*=*{2 + cos

*φ*cos[2π(

*x*+

*y*)

*/D*] + sin

*φ*cos[2π(

*x*−

*y*)

*/D*]}/4 and

*t*(

*x*,

*y*)

*=*{2 + cos(3π

*r*/

*r*

_{0})cos[2π(

*x*+

*y*)

*/D*] + sin(3π

*r*/

*r*

_{0})cos[2π(

*x*−

*y*)

*/D*]}/4, respectively. From the analyzer, the intensity distributions basically meet the theoretical polarization, with some local regions ambiguous.

*s*- and

*p*-components are mirror symmetric with each other. Besides, the wavefront of the beam will be flipped horizontally during each reflection. To make the outputs of

*s*- and

*p*-components have reversed wavefronts, an odd number of mirrors are needed. If, for instance, the trapezoid Sagnac interferometer is replaced by a triangular one, the output field from

*s*- and

*p*-components are exactly the same both for 1D and 2D CGHs, and the superposed field is always linearly polarized. Therefore, trapezoid Sagnac interferometer should be an optimized one for our generating method.

## 4. Conclusion

## Acknowledgments

## References and links

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

2. | K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express |

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

4. | H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics |

5. | X. Jiao, S. Liu, Q. Wang, X. Gan, P. Li, and J. Zhao, “Redistributing energy flow and polarization of a focused azimuthally polarized beam with rotationally symmetric sector-shaped obstacles,” Opt. Lett. |

6. | K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. |

7. | Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. |

8. | X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. |

9. | O. V. Angelsky, A. Y. Bekshaev, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and C. Y. Zenkova, “Orbital rotation without orbital angular momentum: mechanical action of the spin part of the internal energy flow in light beams,” Opt. Express |

10. | Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett. |

11. | H. Wang, L. Shi, G. Yuan, X. Miao, W. Tan, and T. Chong, “Subwavelength and super-resolution nondiffraction beam,” Appl. Phys. Lett. |

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

13. | M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process. |

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

15. | Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. |

16. | 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. |

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

18. | P. B. Phua, W. J. Lai, Y. L. Lim, K. S. Tiaw, B. C. Lim, H. H. Teo, and M. H. Hong, “Mimicking optical activity for generating radially polarized light,” Opt. Lett. |

19. | S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. |

20. | N. Passilly, R. de Saint Denis, K. Aït-Ameur, F. Treussart, R. Hierle, and J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A |

21. | V. G. Niziev, R. S. Chang, and A. V. Nesterov, “Generation of inhomogeneously polarized laser beams by use of a Sagnac interferometer,” Appl. Opt. |

22. | X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. |

23. | C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. |

24. | P. H. Jones, M. Rashid, M. Makita, and O. M. Maragò, “Sagnac interferometer method for synthesis of fractional polarization vortices,” Opt. Lett. |

25. | X. L. Wang, Y. Li, J. Chen, C. S. Guo, J. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express |

26. | H. Chen, J. Hao, B. F. Zhang, J. Xu, J. Ding, and H. T. Wang, “Generation of vector beam with space-variant distribution of both polarization and phase,” Opt. Lett. |

27. | S. Tripathi and K. C. Toussaint Jr., “Versatile generation of optical vector fields and vector beams using a non-interferometric approach,” Opt. Express |

28. | C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express |

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.5790) Instrumentation, measurement, and metrology : Sagnac effect

(260.5430) Physical optics : Polarization

**ToC Category:**

Physical Optics

**History**

Original Manuscript: July 16, 2012

Revised Manuscript: August 19, 2012

Manuscript Accepted: August 19, 2012

Published: September 6, 2012

**Citation**

Sheng Liu, Peng Li, Tao Peng, and Jianlin Zhao, "Generation of arbitrary spatially variant polarization beams with a trapezoid Sagnac interferometer," Opt. Express **20**, 21715-21721 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-19-21715

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### References

- Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon.1(1), 1–57 (2009). [CrossRef]
- K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7(2), 77–87 (2000). [CrossRef] [PubMed]
- R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91(23), 233901 (2003). [CrossRef] [PubMed]
- H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008). [CrossRef]
- X. Jiao, S. Liu, Q. Wang, X. Gan, P. Li, and J. Zhao, “Redistributing energy flow and polarization of a focused azimuthally polarized beam with rotationally symmetric sector-shaped obstacles,” Opt. Lett.37(6), 1041–1043 (2012). [CrossRef] [PubMed]
- K. Y. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett.96(7), 073903 (2006). [CrossRef] [PubMed]
- Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett.99(7), 073901 (2007). [CrossRef] [PubMed]
- X. L. Wang, J. Chen, Y. Li, J. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett.105(25), 253602 (2010). [CrossRef] [PubMed]
- O. V. Angelsky, A. Y. Bekshaev, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and C. Y. Zenkova, “Orbital rotation without orbital angular momentum: mechanical action of the spin part of the internal energy flow in light beams,” Opt. Express20(4), 3563–3571 (2012). [CrossRef] [PubMed]
- Q. Zhan, “Evanescent Bessel beam generation via surface plasmon resonance excitation by a radially polarized beam,” Opt. Lett.31(11), 1726–1728 (2006). [CrossRef] [PubMed]
- H. Wang, L. Shi, G. Yuan, X. Miao, W. Tan, and T. Chong, “Subwavelength and super-resolution nondiffraction beam,” Appl. Phys. Lett.89(17), 171102 (2006). [CrossRef]
- T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett.33(2), 122–124 (2008). [CrossRef] [PubMed]
- M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process.86(3), 329–334 (2007). [CrossRef]
- M. Stalder and M. Schadt, “Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters,” Opt. Lett.21(23), 1948–1950 (1996). [CrossRef] [PubMed]
- Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett.79(11), 1587–1589 (2001). [CrossRef]
- 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(3), 238–240 (2004). [CrossRef] [PubMed]
- G. Machavariani, Y. Lumer, I. Moshe, A. Meir, and S. Jackel, “Efficient extracavity generation of radially and azimuthally polarized beams,” Opt. Lett.32(11), 1468–1470 (2007). [CrossRef] [PubMed]
- P. B. Phua, W. J. Lai, Y. L. Lim, K. S. Tiaw, B. C. Lim, H. H. Teo, and M. H. Hong, “Mimicking optical activity for generating radially polarized light,” Opt. Lett.32(4), 376–378 (2007). [CrossRef] [PubMed]
- S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt.29(15), 2234–2239 (1990). [CrossRef] [PubMed]
- N. Passilly, R. de Saint Denis, K. Aït-Ameur, F. Treussart, R. Hierle, and J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A22(5), 984–991 (2005). [CrossRef] [PubMed]
- V. G. Niziev, R. S. Chang, and A. V. Nesterov, “Generation of inhomogeneously polarized laser beams by use of a Sagnac interferometer,” Appl. Opt.45(33), 8393–8399 (2006). [CrossRef] [PubMed]
- X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett.32(24), 3549–3551 (2007). [CrossRef] [PubMed]
- C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys.9(3), 78 (2007). [CrossRef]
- P. H. Jones, M. Rashid, M. Makita, and O. M. Maragò, “Sagnac interferometer method for synthesis of fractional polarization vortices,” Opt. Lett.34(17), 2560–2562 (2009). [CrossRef] [PubMed]
- X. L. Wang, Y. Li, J. Chen, C. S. Guo, J. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express18(10), 10786–10795 (2010). [CrossRef] [PubMed]
- H. Chen, J. Hao, B. F. Zhang, J. Xu, J. Ding, and H. T. Wang, “Generation of vector beam with space-variant distribution of both polarization and phase,” Opt. Lett.36(16), 3179–3181 (2011). [CrossRef] [PubMed]
- S. Tripathi and K. C. Toussaint., “Versatile generation of optical vector fields and vector beams using a non-interferometric approach,” Opt. Express20(10), 10788–10795 (2012). [CrossRef] [PubMed]
- C. A. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express13(5), 1749–1760 (2005). [CrossRef] [PubMed]

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