## Search for Hermite-Gauss mode rotation in cholesteric liquid crystals |

Optics Express, Vol. 19, Issue 14, pp. 12978-12983 (2011)

http://dx.doi.org/10.1364/OE.19.012978

Acrobat PDF (1703 KB)

### Abstract

In theory, there are analogous transformations of light’s spin and orbital angular momentum [Allen and Padgett, J. Mod. Opt. **54**, 487 (2007)]; however, none have been observed experimentally yet. In particular, it is unknown if there exists for the orbital angular momentum of light an effect analogous to the spin angular momentum-based optical rotation; this would manifest itself as a rotation of the corresponding Hermite-Gauss mode. Here we report an experimental search for this effect in a cholesteric liquid crystal polymer, using strongly focussed, spin-orbit coupled light. We find that the relative phase velocities of the orbital modes constituting the Hermite-Gauss mode agree to within 10^{−5}.

© 2011 OSA

## 1. Introduction

6. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. **75**, 826–829 (1995). [CrossRef] [PubMed]

8. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. **96**, 123–132 (1993). [CrossRef]

6. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. **75**, 826–829 (1995). [CrossRef] [PubMed]

9. S. J. van Enk and G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. **41**, 963–977 (1994). [CrossRef]

16. V. Klimov, D. Bloch, M. Ducloy, and J. R. R. Leite, “Detecting photons in the dark region of Laguerre-Gauss beams,” Opt. Express **17**, 9718–9723 (2009). [CrossRef] [PubMed]

17. F. Araoka, T. Verbiest, K. Clays, and A. Persoons, “Interactions of twisted light with chiral molecules: an experimental investigation,” Phys. Rev. A **71**, 055401 (2005). [CrossRef]

12. R. Jáuregui, “Rotational effects of twisted light on atoms beyond the paraxial approximation,” Phys. Rev. A **70**, 033415 (2004). [CrossRef]

14. A. Alexandrescu, D. Cojoc, and E. D. Fabrizio, “Mechanism of angular momentum exchange between molecules and laguerre-gaussian beams,” Phys. Rev. Lett. **96**, 243001 (2006). [CrossRef] [PubMed]

17. F. Araoka, T. Verbiest, K. Clays, and A. Persoons, “Interactions of twisted light with chiral molecules: an experimental investigation,” Phys. Rev. A **71**, 055401 (2005). [CrossRef]

10. M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. Dávila Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett. **89**, 143601 (2002). [CrossRef] [PubMed]

11. L. C. Dávila Romero, D. L. Andrews, and M. Babiker, “A quantum electrodynamics framework for the nonlinear optics of twisted beams,” J. Opt. B **4**, S66 (2002). [CrossRef]

13. D. L. Andrews, L. C. Dávila Romero, and M. Babiker, “On optical vortex interactions with chiral matter,” Opt. Commun. **237**, 133–139 (2004). [CrossRef]

15. R. Grinter, “Photon angular momentum: selection rules and multipolar transition moments,” J. Phys. B **41**, 095001 (2008). [CrossRef]

18. M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. **97**, 170406 (2006). [CrossRef] [PubMed]

19. D. Sanvitto, F. M. Marchetti, M. H. Szymanska, G. Tosi, M. Baudisch, F. P. Laussy, D. N. Krizhanovskii, M. S. Skolnick, L. Marrucci, A. Lemaitre, J. Bloch, C. Tejedor, and L. Vina, “Persistent currents and quantized vortices in a polariton superfluid,” Nat. Phys. **6**, 527–533 (2010). [CrossRef]

20. A. Picón, J. Mompart, J. R. V. de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Photoionization with orbital angular momentum beams,” Opt. Express **18**, 3660–3671 (2010). [CrossRef] [PubMed]

*per se*sensitive to the handedness of the OAM, which is key to find the OAM analogue to optical activity.

21. I. Khoo, *Liquid Crystals*, Wiley series in pure and applied optics (Wiley-Interscience, 2007). [CrossRef]

*ℓ*= ±1 show similar symmetry.

22. W. D. St. John, W. J. Fritz, Z. J. Lu, and D.-K. Yang, “Bragg reflection from cholesteric liquid crystals,” Phys. Rev. E **51**, 1191–1198 (1995). [CrossRef]

24. M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. **73**, 096501 (2010). [CrossRef]

*λ*, but from multiple reflection, retardation, and interference of light. Well-known effects in this category in photonic crystals are the superprism effect [25

25. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B **58**, R10096–R10099 (1998). [CrossRef]

26. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature **423**, 604–605 (2003). [CrossRef] [PubMed]

2. L. Allen and M. Padgett, “Equivalent geometric transformations for spin and orbital angular momentum of light,” J. Mod. Opt. **54**, 487–491 (2007). [CrossRef]

*HG*

_{1,0}Hermite-Gaussian beam [2

2. L. Allen and M. Padgett, “Equivalent geometric transformations for spin and orbital angular momentum of light,” J. Mod. Opt. **54**, 487–491 (2007). [CrossRef]

*ϕ*depends on the relative phase of its components in the LG mode decomposition: If we introduce in a HG beam a (hypothetical) medium with different effective refractive indices

*n*

_{+}

*and*

_{ℓ}*n*

_{–}

*of the*

_{ℓ}*LG*

_{–}

_{ℓ}_{,0}and

*LG*

_{+}

_{ℓ}_{,0}modes (

*n*

_{+}

*≠*

_{ℓ}*n*

_{–}

*for*

_{ℓ}*ℓ*= 1), i.e., providing intermodal dispersion of these modes, the HG mode pattern is rotated. This is the closest possible analogue to conventional linear optical activity.

## 2. Experimental setting

*p*

_{0}(a full 2

*π*rotation of the director, see Fig. 1) is of the order of the optical wavelength [27

27. D. J. Broer and I. Heynderickx, “Three-dimensionally ordered polymer networks with a helicoidal structure,” Macromolecules **23**, 2474–2477 (1990). [CrossRef]

28. R. A. M. Hikmet, J. Lub, and D. J. Broer, “Anisotropic networks formed by photopolymerization of liquid-crystalline molecules,” Adv. Mater. **3**, 392–394 (1991). [CrossRef]

*μm*spacers. Photopolymerization was initiated in the chiral nematic phase at 85 °

*C*. In the Bragg-type reflection band [22

22. W. D. St. John, W. J. Fritz, Z. J. Lu, and D.-K. Yang, “Bragg reflection from cholesteric liquid crystals,” Phys. Rev. E **51**, 1191–1198 (1995). [CrossRef]

29. D. W. Berreman and T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid-crystal films,” Phys. Rev. Lett. **25**, 577–581 (1970). [CrossRef]

*λ*

_{0}=

*n̄p*

_{0}(full width Δ

*λ*=

*p*

_{0}Δ

*n*, where Δ

*n*= |

*n*–

_{e}*n*|), the circular polarization component, which has the same handedness as the CLC, is reflected, while the other component is transmitted (see Fig. 1).

_{o}*n̄*= (

*n*+

_{o}*n*)/2 is the average refractive index; for a non-chiral variant of the LC, the refractive indices at

_{e}*λ*= 500

*nm*have been measured to be

*n*≈ 1.55 and

_{o}*n*≈ 1.70. Fig. 1b shows the typical dispersion-like optical rotation of the sample with a reflection band at 550–590 nm. The optical rotation is very large: Close to the reflection band, the optical rotation is around 2.3 × 10

_{e}^{4}degrees/mm, this is very much larger than what is possible in samples with molecular optical activity only. This allows us to use a thin sample and microscopy objectives for focussing.

*J*, normalized to the energy flux per unit length

_{z}*E*is given by [30

30. S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. **110**, 670–678 (1994). [CrossRef]

*σ*is the spin AM,

_{z}*ω*the wave number,

*ℓ*and

*p*the azimuthal and radial index of the Laguerre-Gaussian mode, and

*θ*

_{0}is the half aperture angle of the focussed beam. Eq. (2) follows directly from that in [30

30. S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. **110**, 670–678 (1994). [CrossRef]

*w*

_{0}=

*λ*/(

*πθ*), where

*w*

_{0}and

*z*are the Gaussian beam width and Rayleigh range, respectively. The second term in Eq. (2) is the consequence of spin-orbit coupling, and

_{R}31. T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Angular momentum of a strongly focused gaussian beam,” J. Opt. A **10**, 115005 (2008). [CrossRef]

*NA*= 0.4, this results in

## 3. Results and discussion

*n*

_{+}

*–*

_{ℓ}*n*

_{–}

*| < 8·10*

_{ℓ}^{−6}for

*ℓ*= 1. Probably this limit could be lowered by using a specially designed superposition of higher-order OAM modes, however, our data analysis method works best for the

*ℓ*= ±1 case.

*k⃗*=

_{out}*R ·k⃗*, where

_{in}*R*is a rotation matrix which rotates the wave vector

*k⃗*around the geometric beam axis. If the medium is translation invariant in the sample plane, in linear optics, such a

_{in}*R*can not be constructed. However, this translation symmetry holds only within an effective refractive index model, but not on the atomic scale. Even if the wavelength is much larger than the microscopic length scale of the system (e.g., the crystal lattice constant), the optical properties are influenced by the microscopic structure. This is well known in crystal optics: Within the dipole approximation, cubic crystals should be isotropic, however, many show birefringence due to spatial dispersion. This effect has been predicted by Lorentz in 1878, firstly discovered in [32

32. J. Pastrnak and K. Vedam, “Optical anisotropy of silicon single crystals,” Phys. Rev. B **3**, 2567–2571 (1971). [CrossRef]

33. A. G. Serebryakov and F. Bociort, “Spatial dispersion of crystals as a critical problem for deep uv lithography,” J. Opt. Technol. **70**, 566–569 (2003). [CrossRef]

34. J. W. R. Tabosa and D. V. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. **83**, 4967–4970 (1999). [CrossRef]

18. M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. **97**, 170406 (2006). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. |

2. | L. Allen and M. Padgett, “Equivalent geometric transformations for spin and orbital angular momentum of light,” J. Mod. Opt. |

3. | G. Nienhuis, J. P. Woerdman, and I. Kuščer, “Magnetic and mechanical faraday effects,” Phys. Rev. A |

4. | M. Padgett, G. Whyte, J. Girkin, A. Wright, L. Allen, P. Öhberg, and S. M. Barnett, “Polarization and image rotation induced by a rotating dielectric rod: an optical angular momentum interpretation,” Opt. Lett. |

5. | J. Leach, A. J. Wright, J. B. Götte, J. M. Girkin, L. Allen, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “‘aether drag’ and moving images,” Phys. Rev. Lett. |

6. | H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. |

7. | N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. |

8. | M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. |

9. | S. J. van Enk and G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. |

10. | M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. Dávila Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett. |

11. | L. C. Dávila Romero, D. L. Andrews, and M. Babiker, “A quantum electrodynamics framework for the nonlinear optics of twisted beams,” J. Opt. B |

12. | R. Jáuregui, “Rotational effects of twisted light on atoms beyond the paraxial approximation,” Phys. Rev. A |

13. | D. L. Andrews, L. C. Dávila Romero, and M. Babiker, “On optical vortex interactions with chiral matter,” Opt. Commun. |

14. | A. Alexandrescu, D. Cojoc, and E. D. Fabrizio, “Mechanism of angular momentum exchange between molecules and laguerre-gaussian beams,” Phys. Rev. Lett. |

15. | R. Grinter, “Photon angular momentum: selection rules and multipolar transition moments,” J. Phys. B |

16. | V. Klimov, D. Bloch, M. Ducloy, and J. R. R. Leite, “Detecting photons in the dark region of Laguerre-Gauss beams,” Opt. Express |

17. | F. Araoka, T. Verbiest, K. Clays, and A. Persoons, “Interactions of twisted light with chiral molecules: an experimental investigation,” Phys. Rev. A |

18. | M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. |

19. | D. Sanvitto, F. M. Marchetti, M. H. Szymanska, G. Tosi, M. Baudisch, F. P. Laussy, D. N. Krizhanovskii, M. S. Skolnick, L. Marrucci, A. Lemaitre, J. Bloch, C. Tejedor, and L. Vina, “Persistent currents and quantized vortices in a polariton superfluid,” Nat. Phys. |

20. | A. Picón, J. Mompart, J. R. V. de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Photoionization with orbital angular momentum beams,” Opt. Express |

21. | I. Khoo, |

22. | W. D. St. John, W. J. Fritz, Z. J. Lu, and D.-K. Yang, “Bragg reflection from cholesteric liquid crystals,” Phys. Rev. E |

23. | L. D. Landau and E. M. Lifshitz, |

24. | M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. |

25. | H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B |

26. | E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature |

27. | D. J. Broer and I. Heynderickx, “Three-dimensionally ordered polymer networks with a helicoidal structure,” Macromolecules |

28. | R. A. M. Hikmet, J. Lub, and D. J. Broer, “Anisotropic networks formed by photopolymerization of liquid-crystalline molecules,” Adv. Mater. |

29. | D. W. Berreman and T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid-crystal films,” Phys. Rev. Lett. |

30. | S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. |

31. | T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Angular momentum of a strongly focused gaussian beam,” J. Opt. A |

32. | J. Pastrnak and K. Vedam, “Optical anisotropy of silicon single crystals,” Phys. Rev. B |

33. | A. G. Serebryakov and F. Bociort, “Spatial dispersion of crystals as a critical problem for deep uv lithography,” J. Opt. Technol. |

34. | J. W. R. Tabosa and D. V. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. |

**OCIS Codes**

(030.4070) Coherence and statistical optics : Modes

(160.1585) Materials : Chiral media

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

**ToC Category:**

Materials

**History**

Original Manuscript: April 13, 2011

Revised Manuscript: May 27, 2011

Manuscript Accepted: June 2, 2011

Published: June 21, 2011

**Citation**

W. Löffler, M. P. van Exter, G. W. ’t Hooft, G. Nienhuis, D. J. Broer, and J. P. Woerdman, "Search for Hermite-Gauss mode rotation in cholesteric liquid crystals," Opt. Express **19**, 12978-12983 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-12978

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### References

- M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24, 430–432 (1999). [CrossRef]
- L. Allen and M. Padgett, “Equivalent geometric transformations for spin and orbital angular momentum of light,” J. Mod. Opt. 54, 487–491 (2007). [CrossRef]
- G. Nienhuis, J. P. Woerdman, and I. Kuščer, “Magnetic and mechanical faraday effects,” Phys. Rev. A 46, 7079–7092 (1992). [CrossRef] [PubMed]
- M. Padgett, G. Whyte, J. Girkin, A. Wright, L. Allen, P. Öhberg, and S. M. Barnett, “Polarization and image rotation induced by a rotating dielectric rod: an optical angular momentum interpretation,” Opt. Lett. 31, 2205–2207 (2006). [CrossRef] [PubMed]
- J. Leach, A. J. Wright, J. B. Götte, J. M. Girkin, L. Allen, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “‘aether drag’ and moving images,” Phys. Rev. Lett. 100, 153902 (2008). [CrossRef] [PubMed]
- H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75, 826–829 (1995). [CrossRef] [PubMed]
- N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22, 52–54 (1997). [CrossRef] [PubMed]
- M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]
- S. J. van Enk and G. Nienhuis, “Commutation rules and eigenvalues of spin and orbital angular momentum of radiation fields,” J. Mod. Opt. 41, 963–977 (1994). [CrossRef]
- M. Babiker, C. R. Bennett, D. L. Andrews, and L. C. Dávila Romero, “Orbital angular momentum exchange in the interaction of twisted light with molecules,” Phys. Rev. Lett. 89, 143601 (2002). [CrossRef] [PubMed]
- L. C. Dávila Romero, D. L. Andrews, and M. Babiker, “A quantum electrodynamics framework for the nonlinear optics of twisted beams,” J. Opt. B 4, S66 (2002). [CrossRef]
- R. Jáuregui, “Rotational effects of twisted light on atoms beyond the paraxial approximation,” Phys. Rev. A 70, 033415 (2004). [CrossRef]
- D. L. Andrews, L. C. Dávila Romero, and M. Babiker, “On optical vortex interactions with chiral matter,” Opt. Commun. 237, 133–139 (2004). [CrossRef]
- A. Alexandrescu, D. Cojoc, and E. D. Fabrizio, “Mechanism of angular momentum exchange between molecules and laguerre-gaussian beams,” Phys. Rev. Lett. 96, 243001 (2006). [CrossRef] [PubMed]
- R. Grinter, “Photon angular momentum: selection rules and multipolar transition moments,” J. Phys. B 41, 095001 (2008). [CrossRef]
- V. Klimov, D. Bloch, M. Ducloy, and J. R. R. Leite, “Detecting photons in the dark region of Laguerre-Gauss beams,” Opt. Express 17, 9718–9723 (2009). [CrossRef] [PubMed]
- F. Araoka, T. Verbiest, K. Clays, and A. Persoons, “Interactions of twisted light with chiral molecules: an experimental investigation,” Phys. Rev. A 71, 055401 (2005). [CrossRef]
- M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97, 170406 (2006). [CrossRef] [PubMed]
- D. Sanvitto, F. M. Marchetti, M. H. Szymanska, G. Tosi, M. Baudisch, F. P. Laussy, D. N. Krizhanovskii, M. S. Skolnick, L. Marrucci, A. Lemaitre, J. Bloch, C. Tejedor, and L. Vina, “Persistent currents and quantized vortices in a polariton superfluid,” Nat. Phys. 6, 527–533 (2010). [CrossRef]
- A. Picón, J. Mompart, J. R. V. de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Photoionization with orbital angular momentum beams,” Opt. Express 18, 3660–3671 (2010). [CrossRef] [PubMed]
- I. Khoo, Liquid Crystals , Wiley series in pure and applied optics (Wiley-Interscience, 2007). [CrossRef]
- W. D. St. John, W. J. Fritz, Z. J. Lu, and D.-K. Yang, “Bragg reflection from cholesteric liquid crystals,” Phys. Rev. E 51, 1191–1198 (1995). [CrossRef]
- L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon Press, 1975), Vol. 8.
- M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. 73, 096501 (2010). [CrossRef]
- H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096–R10099 (1998). [CrossRef]
- E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423, 604–605 (2003). [CrossRef] [PubMed]
- D. J. Broer and I. Heynderickx, “Three-dimensionally ordered polymer networks with a helicoidal structure,” Macromolecules 23, 2474–2477 (1990). [CrossRef]
- R. A. M. Hikmet, J. Lub, and D. J. Broer, “Anisotropic networks formed by photopolymerization of liquid-crystalline molecules,” Adv. Mater. 3, 392–394 (1991). [CrossRef]
- D. W. Berreman and T. J. Scheffer, “Bragg reflection of light from single-domain cholesteric liquid-crystal films,” Phys. Rev. Lett. 25, 577–581 (1970). [CrossRef]
- S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994). [CrossRef]
- T. A. Nieminen, A. B. Stilgoe, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Angular momentum of a strongly focused gaussian beam,” J. Opt. A 10, 115005 (2008). [CrossRef]
- J. Pastrnak and K. Vedam, “Optical anisotropy of silicon single crystals,” Phys. Rev. B 3, 2567–2571 (1971). [CrossRef]
- A. G. Serebryakov and F. Bociort, “Spatial dispersion of crystals as a critical problem for deep uv lithography,” J. Opt. Technol. 70, 566–569 (2003). [CrossRef]
- J. W. R. Tabosa and D. V. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,” Phys. Rev. Lett. 83, 4967–4970 (1999). [CrossRef]

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