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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 21 — Oct. 8, 2012
  • pp: 23138–23143
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Periodic and aperiodic liquid crystal-polymer composite structures realized via spatial light modulator direct holography

M. Infusino, A. De Luca, V. Barna, R. Caputo, and C. Umeton  »View Author Affiliations


Optics Express, Vol. 20, Issue 21, pp. 23138-23143 (2012)
http://dx.doi.org/10.1364/OE.20.023138


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Abstract

In this work we present the first realization and characterization of two-dimensional periodic and aperiodic POLICRYPS (Polymer Liquid Crystal Polymer Slices) structures, obtained by means of a single-beam holographic technique exploiting a high resolution spatial light modulator (SLM). A first investigation shows that the gratings, operating in the Raman Nath regime, exhibit a morphology and a electro-optical behavior that are typical of the POLICRYPS gratings realized by two-beam interference holography.

© 2012 OSA

1. Introduction

2. Method and set-up

SLMs have been extensively exploited for holographic writing purposes [12

12. G. Zito, B. Piccirillo, E. Santamato, A. Marino, V. Tkachenko, and G. Abbate, “Two-dimensional photonic quasi-crystals by single beam computer-generated holography,” Opt. Express 16, 5164–5170 (2008). [CrossRef] [PubMed]

, 14

14. J. Li, Y. Liu, X. Xie, P. Zhang, B. Liang, L. Yan, J. Zhou, G. Kurizki, D. Jacobs, K. S. Wong, and Y. Zhong, “Fabrication of photonic crystals with functional defects by one-step holographic lithography,” Opt. Express 16, 12899–12904 (2008). [CrossRef] [PubMed]

, 15

15. A. Ogiwara and T. Hirokari, “Formation of anisotropic diffraction gratings in a polymer-dispersed liquid crystal by polarization modulation using a spatial light modulator,” Appl. Opt. 47, 3015–3022 (2008). [CrossRef] [PubMed]

]. Methods that allow imaging of the desired light pattern in the object plane are usually referred to as “direct”, in contrast with “indirect” ones which allow the image reconstruction in the Fourier plane. In literature, some examples of direct imaging techniques exploitable by using SLMs are reported [12

12. G. Zito, B. Piccirillo, E. Santamato, A. Marino, V. Tkachenko, and G. Abbate, “Two-dimensional photonic quasi-crystals by single beam computer-generated holography,” Opt. Express 16, 5164–5170 (2008). [CrossRef] [PubMed]

, 16

16. J. A. Davis, K. O. Valadéz, and D. M. Cottrell, “Encoding amplitude and phase information onto a binary phase-only spatial light modulator,” Appl. Opt. 42, 2003–2008 (2003). [CrossRef] [PubMed]

]. The direct imaging method used in the present work is based on the solution proposed by Davis in 1989 [17

17. J. A. Davis, S. W. Flowers, D. M. Cottrell, and R. A. Lilly, “Smoothing of edge-enhanced impulse response from binary phase-only filters using random binary patterns,” Appl. Opt. 28, 2987–2988 (1989). [CrossRef] [PubMed]

] to solve the problem of edge-enhanced response from a binary SLM. The technique allows the realization of any generic binary pattern, characterized only by two intensity levels. The images, loaded on the SLM display, are represented by areas with pixels of the same color (white for example) alternated to areas made of a random distribution of gray levels. Monochromatic and randomized areas in the input plane (SLM) produce bright and dark areas in the output plane respectively. In our case, in order to magnify the obtained pattern to the desired size a 4f Fourier lenses system has been set-up (Fig. 1). The beam, coming from a Nd:YAG laser doubled in frequency (λ = 532 nm), is spatially filtered, enlarged and collimated by the lens L1; then, it is used to illuminate the SLM, which modifies the wavefront. Finally, the light pattern coming from the SLM is magnified and relayed to the sample plane by the two lenses L2 and L3. The magnification ratio M = f3/f2 depends on the focal lengths f2 and f3 of the lenses L2 and L3 respectively. A photo-sensitive pre-polymer syrup, made of the pre-polymer system NOA61 (70–72% wt, by Norland), the Nematic Liquid Crystal E7 (28–30% wt by Merck) and the photo-initiator Irgacure 784 (1–2% wt by BASF Resins) is used to fill in, by capillarity, the sample cell, obtained by putting two glass substrates at a controlled distance. The choice of this mixture, formulated by Natarajan et al. [18

18. L. V. Natarajan, C. K. Shepherd, D. M. Brandelik, R. L. Sutherland, S. Chandra, V. P. Tondiglia, D. Tomlin, and T. J. Bunning, “Switchable holographic polymer-dispersed liquid crystal reflection gratings based on thiolene photopolymerization,” Chem. Mater. 15, 2477–2484, (2003). [CrossRef]

] and very similar to the one previously used in reference [19

19. M. E. De Rosa, V. P. Tondiglia, and L. V. Natarajan, “Mechanical deformation of a liquid crystal diffraction grating in an elastic polymer,” J. Appl. Polym. Sci. 68, 523–526 (1998). [CrossRef]

], is the result of a dedicated investigation performed to extend the use of POLICRYPS technology also to systems exploiting visible-light curing sources; the detailed study of the attempts that brought to the determination of this chemical composition and the obtained results are reported elsewhere [20

20. M. Infusino, A. Ferraro, A. De Luca, R. Caputo, and C. Umeton, “Policryps visible curing for spatial light modulator based holography,” submitted J. Opt. Soc. Am. B, (2012).

]. In our experiments, the exploitation of the POLICRYPS technology consisted in exposing the sample to the light pattern produced by the SLM at a temperature higher than the Nematic-Isotropic transition point. It is worth noting that the NLC concentration in the pre-polymer mixture has a solubility threshold of about 30%wt; when this value is overcome, NLC droplets appear during the polymerization process, thus affecting the typical POLICRYPS morphology. Polymeric branches being formed in correspondence of the bright areas of the curing pattern [21

21. A. Veltri, R. Caputo, C. Umeton, and A. V. Sukhov, “Model for the photoinduced formation of diffraction gratings in liquid-crystalline composite materials,” Appl. Phys. Lett. 84, 3492–3494 (2004). [CrossRef]

], it is convenient to adjust, a priori, the ratio between bright and dark areas as the one corresponding to the maximum achievable phase separation (70:30); the possibility of doing this kind of choice represents an innovative advantage of using a SLM for fabricating POLICRYPS structures.

Fig. 1 4f Fourier set-up for image reconstruction

3. Result and discussion

Fig. 2 (a) Optical microscopy image of a 2D POLICRYPS grid with a pitch of 38 μm. The inset shows the presence of disclinations in the region of intersection between vertical and horizontal stripes; (b) Far-field diffraction pattern for red and green light probes. (c) Polar graphs of the (1,0) and (0,1) orders; (d) Optical response of the sample to an applied (ON-OFF) electric stimulus.

The main benefit of the new exploited technique is represented by the possibility to realize asymmetric geometries, hardly achievable by using even a multi-beam interference technique. In this framework, the fork grating is one of the most representative examples. This is a diffractive optical element whose diffraction pattern is composed by several orders, each of them being an optical vortex (Fig. 3(c)). A vortex is a light beam characterized by an helical wavefront described by the phase function ψ1 = exp(iqθ), where θ is the azimuthal angle of a cylindrical coordinate system (r, θ, z) around the z axis, which indicates the beam propagation direction. For points belonging to the helical axis (r = 0), the phase is not defined and the corresponding field amplitude is zero: an optical vortex projected on a screen appears therefore like a ring with a zero intensity region in the center (Fig. 3(d)). The integer value q that appears in the ψ1 expression indicates the number of phase winding around the dark spot.

Fig. 3 (a) Optical microscopy image of a fork grating with a pitch of 13μm. (b) Switching curves for s- (red squares) and p-polarized (green squares) incoming light. (c) Far-field diffraction pattern for a red probe. (d) Beam-profiler acquisition of the first diffracted order and (e) the related cut along a radius (θ = π/2)

Optical vortexes are widely used for optical trapping of microscopic dielectric particles. Indeed, in conventional optical trapping, a Gaussian laser beam can trap those particles whose refractive index is greater than that of the surrounding medium. On the contrary, due to the gradient of intensity with the dark center, optical vortexes can trap particles whose refractive index value is both higher or lower than the one of the surrounding medium [22

22. K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996). [CrossRef] [PubMed]

,23

23. K. T. Gahagan and G. A. Swartzlander Jr., “Trapping of low-index microparticles in an optical vortex,” J. Opt. Soc. Am. B 15, 524–534 (1998). [CrossRef]

]. We reckon, therefore, that the realization of cheap devices endowed with POLICRYPS electro-optical characteristics and able to produce switchable optical vortexes represents a quite interesting application [24

24. Y. J. Liu, X. W. Sun, Q. Wang, and D. Luo, “Electrically switchable optical vortex generated by computer-generated hologram recorded in polymer-dispersed liquid crystals,” Opt. Express 15, 16645–16650 (2007). [CrossRef] [PubMed]

]. The pattern designing the fork grating has been calculated via computer generated holography [25

25. A. Kumar, P. Vaity, Y. Krishna, and R. P. Singh, “Engineering the size of dark core of an optical vortex,” Opt. Lasers Eng. 48, 276–281 (2010). [CrossRef]

, 26

26. A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. 76, 916–921 (2008). [CrossRef]

]. More precisely, it has been obtained by plotting the intensity distribution Ifork calculated as the interference of two waves: the object beam containing the vortex phase ψ1 and a plane-wave reference beam with phase ψ2.
Ifork=|ψ1+ψ2|2=|exp(iqθ)+exp(ikz)|2=2[1cos(kzqθ)]
(1)

4. Conclusions

Standard holographic techniques can reveal quite complicate and troublesome or, eventually, not suitable when the aim is the fabrication of 2D periodic or aperiodic structures. In order to face this issue, we combined the excellent morphology and features achievable by exploiting the POLICRYPS structure with the freedom in design offered by the utilization of a SLM. In this way, we obtained a quick and low-cost one-step procedure that allows the fabrication of a large variety of high quality structures. Moreover, the technique, used can be further developed in order to realize diffractive structures operating in the Bragg regime. In fact, despite SLM limited resolution, smaller periodicities up to few μm can be achieved.

Acknowledgment

The research work was supported from the Romanian national projects CNCSIS TE 225-96/2010 and ID-PCE-2011/3/1007.

References and links

1.

R. Caputo, L. De Sio, A. Veltri, C. Umeton, and A. V. Sukhov, “Development of a new kind of switchable holographic grating made of liquid-crystal films separated by slices of polymeric material,” Opt. Lett. 29, 1261–1263 (2004). [CrossRef] [PubMed]

2.

R. Caputo, L. De Sio, A. Veltri, C. Umeton, and A. V. Sukhov, “Policryps switchable holographic grating: a promising grating electro-optical pixel for high resolution display application,” J. Disp. Technol. 2, 38–51 (2006). [CrossRef]

3.

R. L. Sutherland, L. V. Natarajan, V. P. Tondiglia, and T. J. Bunning, “Bragg gratings in an acrylate polymer consisting of periodic polymer-dispersed liquid-crystal planes,” Chem. Mater. 5, 1533–1538 (1993). [CrossRef]

4.

R. L. Sutherland, V. P. Tondiglia, L. V. Natarajan, T. J. Bunning, and W. W. Adams, “Electrically switchable volume gratings in polymer-dispersed liquid crystals,” Appl. Phys. Lett. 64, 1074–1076 (1994). [CrossRef]

5.

T. J. White, L. V. Natarajan, V. P. Tondiglia, P. F. Lloyd, T. J. Bunning, and C. A. Guymon, “Monomer functionality effects in the formation of thiol-ene holographic polymer dispersed liquid crystals,” Macromolecules 40, 1121–1127 (2007). [CrossRef]

6.

R. T. Pogue, R. L. Sutherland, M. G. Schmitt, L. V. Natarajan, S. A. Siwecki, V. P. Tondiglia, and T. J. Bunning, “Electrically switchable Bragg gratings from liquid crystal/polymer composites,” Appl. Spectrosc. 54, 12A–28A, (2000). [CrossRef]

7.

A. d’Alessandro, R. Asquini, C. Gizzi, R. Caputo, C. Umeton, A. Veltri, and A. V. Sukhov, “Electro-optical properties of switchable gratings made of polymer and nematic liquid-crystal slices,” Opt. Lett. 29, 1405–1407 (2004). [CrossRef]

8.

R. L. Sutherland, V. P. Tondiglia, L. V. Natarajan, S. Chandra, D. Tomlin, and T. J. Bunning, “Switchable orthorhombic F photonic crystals formed by holographic polymerization-induced phase separation of liquid crystal,” Opt. Express 10, 1074–1082 (2002). [PubMed]

9.

V. P. Tondiglia, L. V. Natarajan, R. L. Sutherland, D. Tomlin, and T. J. Bunning, “Holographic formation of electro-optical polymer-liquid crystal photonic crystals,” Adv. Mater. 14, 187–191 (2002). [CrossRef]

10.

Y. J. Liu and X. W. Sun, “Electrically tunable two-dimensional holographic photonic crystal fabricated by a single diffractive element,” Appl. Phys. Lett. 89, 171101–171103 (2006). [CrossRef]

11.

G. Zito, B. Piccirillo, E. Santamato, A. Marino, V. Tkachenko, and G. Abbate, “Computer-generated holographic gratings in soft matter,” Mol. Cryst. Liq. Cryst. 465, 371–378 (2007). [CrossRef]

12.

G. Zito, B. Piccirillo, E. Santamato, A. Marino, V. Tkachenko, and G. Abbate, “Two-dimensional photonic quasi-crystals by single beam computer-generated holography,” Opt. Express 16, 5164–5170 (2008). [CrossRef] [PubMed]

13.

L. De Sio and C. Umeton, “Dual-mode control of light by two-dimensional periodic structures realized in liquid-crystalline composite materials,” Opt. Lett. 35, 2759–2761 (2010). [CrossRef] [PubMed]

14.

J. Li, Y. Liu, X. Xie, P. Zhang, B. Liang, L. Yan, J. Zhou, G. Kurizki, D. Jacobs, K. S. Wong, and Y. Zhong, “Fabrication of photonic crystals with functional defects by one-step holographic lithography,” Opt. Express 16, 12899–12904 (2008). [CrossRef] [PubMed]

15.

A. Ogiwara and T. Hirokari, “Formation of anisotropic diffraction gratings in a polymer-dispersed liquid crystal by polarization modulation using a spatial light modulator,” Appl. Opt. 47, 3015–3022 (2008). [CrossRef] [PubMed]

16.

J. A. Davis, K. O. Valadéz, and D. M. Cottrell, “Encoding amplitude and phase information onto a binary phase-only spatial light modulator,” Appl. Opt. 42, 2003–2008 (2003). [CrossRef] [PubMed]

17.

J. A. Davis, S. W. Flowers, D. M. Cottrell, and R. A. Lilly, “Smoothing of edge-enhanced impulse response from binary phase-only filters using random binary patterns,” Appl. Opt. 28, 2987–2988 (1989). [CrossRef] [PubMed]

18.

L. V. Natarajan, C. K. Shepherd, D. M. Brandelik, R. L. Sutherland, S. Chandra, V. P. Tondiglia, D. Tomlin, and T. J. Bunning, “Switchable holographic polymer-dispersed liquid crystal reflection gratings based on thiolene photopolymerization,” Chem. Mater. 15, 2477–2484, (2003). [CrossRef]

19.

M. E. De Rosa, V. P. Tondiglia, and L. V. Natarajan, “Mechanical deformation of a liquid crystal diffraction grating in an elastic polymer,” J. Appl. Polym. Sci. 68, 523–526 (1998). [CrossRef]

20.

M. Infusino, A. Ferraro, A. De Luca, R. Caputo, and C. Umeton, “Policryps visible curing for spatial light modulator based holography,” submitted J. Opt. Soc. Am. B, (2012).

21.

A. Veltri, R. Caputo, C. Umeton, and A. V. Sukhov, “Model for the photoinduced formation of diffraction gratings in liquid-crystalline composite materials,” Appl. Phys. Lett. 84, 3492–3494 (2004). [CrossRef]

22.

K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996). [CrossRef] [PubMed]

23.

K. T. Gahagan and G. A. Swartzlander Jr., “Trapping of low-index microparticles in an optical vortex,” J. Opt. Soc. Am. B 15, 524–534 (1998). [CrossRef]

24.

Y. J. Liu, X. W. Sun, Q. Wang, and D. Luo, “Electrically switchable optical vortex generated by computer-generated hologram recorded in polymer-dispersed liquid crystals,” Opt. Express 15, 16645–16650 (2007). [CrossRef] [PubMed]

25.

A. Kumar, P. Vaity, Y. Krishna, and R. P. Singh, “Engineering the size of dark core of an optical vortex,” Opt. Lasers Eng. 48, 276–281 (2010). [CrossRef]

26.

A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys. 76, 916–921 (2008). [CrossRef]

OCIS Codes
(050.1970) Diffraction and gratings : Diffractive optics
(160.3710) Materials : Liquid crystals
(160.5470) Materials : Polymers
(070.6120) Fourier optics and signal processing : Spatial light modulators

ToC Category:
Diffraction and Gratings

History
Original Manuscript: May 30, 2012
Revised Manuscript: August 27, 2012
Manuscript Accepted: August 27, 2012
Published: September 24, 2012

Citation
M. Infusino, A. De Luca, V. Barna, R. Caputo, and C. Umeton, "Periodic and aperiodic liquid crystal-polymer composite structures realized via spatial light modulator direct holography," Opt. Express 20, 23138-23143 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-21-23138


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References

  1. R. Caputo, L. De Sio, A. Veltri, C. Umeton, and A. V. Sukhov, “Development of a new kind of switchable holographic grating made of liquid-crystal films separated by slices of polymeric material,” Opt. Lett.29, 1261–1263 (2004). [CrossRef] [PubMed]
  2. R. Caputo, L. De Sio, A. Veltri, C. Umeton, and A. V. Sukhov, “Policryps switchable holographic grating: a promising grating electro-optical pixel for high resolution display application,” J. Disp. Technol.2, 38–51 (2006). [CrossRef]
  3. R. L. Sutherland, L. V. Natarajan, V. P. Tondiglia, and T. J. Bunning, “Bragg gratings in an acrylate polymer consisting of periodic polymer-dispersed liquid-crystal planes,” Chem. Mater.5, 1533–1538 (1993). [CrossRef]
  4. R. L. Sutherland, V. P. Tondiglia, L. V. Natarajan, T. J. Bunning, and W. W. Adams, “Electrically switchable volume gratings in polymer-dispersed liquid crystals,” Appl. Phys. Lett.64, 1074–1076 (1994). [CrossRef]
  5. T. J. White, L. V. Natarajan, V. P. Tondiglia, P. F. Lloyd, T. J. Bunning, and C. A. Guymon, “Monomer functionality effects in the formation of thiol-ene holographic polymer dispersed liquid crystals,” Macromolecules40, 1121–1127 (2007). [CrossRef]
  6. R. T. Pogue, R. L. Sutherland, M. G. Schmitt, L. V. Natarajan, S. A. Siwecki, V. P. Tondiglia, and T. J. Bunning, “Electrically switchable Bragg gratings from liquid crystal/polymer composites,” Appl. Spectrosc.54, 12A–28A, (2000). [CrossRef]
  7. A. d’Alessandro, R. Asquini, C. Gizzi, R. Caputo, C. Umeton, A. Veltri, and A. V. Sukhov, “Electro-optical properties of switchable gratings made of polymer and nematic liquid-crystal slices,” Opt. Lett.29, 1405–1407 (2004). [CrossRef]
  8. R. L. Sutherland, V. P. Tondiglia, L. V. Natarajan, S. Chandra, D. Tomlin, and T. J. Bunning, “Switchable orthorhombic F photonic crystals formed by holographic polymerization-induced phase separation of liquid crystal,” Opt. Express10, 1074–1082 (2002). [PubMed]
  9. V. P. Tondiglia, L. V. Natarajan, R. L. Sutherland, D. Tomlin, and T. J. Bunning, “Holographic formation of electro-optical polymer-liquid crystal photonic crystals,” Adv. Mater.14, 187–191 (2002). [CrossRef]
  10. Y. J. Liu and X. W. Sun, “Electrically tunable two-dimensional holographic photonic crystal fabricated by a single diffractive element,” Appl. Phys. Lett.89, 171101–171103 (2006). [CrossRef]
  11. G. Zito, B. Piccirillo, E. Santamato, A. Marino, V. Tkachenko, and G. Abbate, “Computer-generated holographic gratings in soft matter,” Mol. Cryst. Liq. Cryst.465, 371–378 (2007). [CrossRef]
  12. G. Zito, B. Piccirillo, E. Santamato, A. Marino, V. Tkachenko, and G. Abbate, “Two-dimensional photonic quasi-crystals by single beam computer-generated holography,” Opt. Express16, 5164–5170 (2008). [CrossRef] [PubMed]
  13. L. De Sio and C. Umeton, “Dual-mode control of light by two-dimensional periodic structures realized in liquid-crystalline composite materials,” Opt. Lett.35, 2759–2761 (2010). [CrossRef] [PubMed]
  14. J. Li, Y. Liu, X. Xie, P. Zhang, B. Liang, L. Yan, J. Zhou, G. Kurizki, D. Jacobs, K. S. Wong, and Y. Zhong, “Fabrication of photonic crystals with functional defects by one-step holographic lithography,” Opt. Express16, 12899–12904 (2008). [CrossRef] [PubMed]
  15. A. Ogiwara and T. Hirokari, “Formation of anisotropic diffraction gratings in a polymer-dispersed liquid crystal by polarization modulation using a spatial light modulator,” Appl. Opt.47, 3015–3022 (2008). [CrossRef] [PubMed]
  16. J. A. Davis, K. O. Valadéz, and D. M. Cottrell, “Encoding amplitude and phase information onto a binary phase-only spatial light modulator,” Appl. Opt.42, 2003–2008 (2003). [CrossRef] [PubMed]
  17. J. A. Davis, S. W. Flowers, D. M. Cottrell, and R. A. Lilly, “Smoothing of edge-enhanced impulse response from binary phase-only filters using random binary patterns,” Appl. Opt.28, 2987–2988 (1989). [CrossRef] [PubMed]
  18. L. V. Natarajan, C. K. Shepherd, D. M. Brandelik, R. L. Sutherland, S. Chandra, V. P. Tondiglia, D. Tomlin, and T. J. Bunning, “Switchable holographic polymer-dispersed liquid crystal reflection gratings based on thiolene photopolymerization,” Chem. Mater.15, 2477–2484, (2003). [CrossRef]
  19. M. E. De Rosa, V. P. Tondiglia, and L. V. Natarajan, “Mechanical deformation of a liquid crystal diffraction grating in an elastic polymer,” J. Appl. Polym. Sci.68, 523–526 (1998). [CrossRef]
  20. M. Infusino, A. Ferraro, A. De Luca, R. Caputo, and C. Umeton, “Policryps visible curing for spatial light modulator based holography,” submitted J. Opt. Soc. Am. B, (2012).
  21. A. Veltri, R. Caputo, C. Umeton, and A. V. Sukhov, “Model for the photoinduced formation of diffraction gratings in liquid-crystalline composite materials,” Appl. Phys. Lett.84, 3492–3494 (2004). [CrossRef]
  22. K. T. Gahagan and G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett.21, 827–829 (1996). [CrossRef] [PubMed]
  23. K. T. Gahagan and G. A. Swartzlander, “Trapping of low-index microparticles in an optical vortex,” J. Opt. Soc. Am. B15, 524–534 (1998). [CrossRef]
  24. Y. J. Liu, X. W. Sun, Q. Wang, and D. Luo, “Electrically switchable optical vortex generated by computer-generated hologram recorded in polymer-dispersed liquid crystals,” Opt. Express15, 16645–16650 (2007). [CrossRef] [PubMed]
  25. A. Kumar, P. Vaity, Y. Krishna, and R. P. Singh, “Engineering the size of dark core of an optical vortex,” Opt. Lasers Eng.48, 276–281 (2010). [CrossRef]
  26. A. V. Carpentier, H. Michinel, J. R. Salgueiro, and D. Olivieri, “Making optical vortices with computer-generated holograms,” Am. J. Phys.76, 916–921 (2008). [CrossRef]

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