## Generation of continuously tunable fractional optical orbital angular momentum using internal conical diffraction |

Optics Express, Vol. 18, Issue 16, pp. 16480-16485 (2010)

http://dx.doi.org/10.1364/OE.18.016480

Acrobat PDF (901 KB)

### Abstract

When a left-circularly polarised Gaussian light beam, which has spin angular momentum (SAM) *J*_{sp} = *σ*ħ = 1ħ per photon, is incident along one of the optic axes of a slab of biaxial crystal it undergoes internal conical diffraction and propagates as a hollow cone of light in the crystal. The emergent beam is a superposition of equal amplitude zero and first order Bessel like beams. The zero order beam is left-circularly polarised with zero orbital angular momentum (OAM) *J*_{orb} = *ℓ*ħ = 0, while the first order beam is right-circularly polarized but carries OAM of *J*_{orb} = 1ħ per photon. Thus, taken together the two beams have zero SAM and *J*_{orb} = ½ħ per photon. In this paper we examine internal conical diffraction of an elliptically polarised beam, which has fractional SAM, and demonstrate an all-optical process for the generation light beams with fractional OAM up to ± 1ħ

© 2010 OSA

## 1. Introduction

*θ*is the azimuthal angle measured around the beam axis and

*ℓ*is the azimuthal mode index [1–3

3. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. **46**(1), 15–28 (2005). [CrossRef]

*ℓ*, in which the phase winds by 2π

*ℓ*on a closed path around the beam. Experimentally, beams with OAM can be generated via numerous methods such as computer-generated fork holograms [4

4. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer-generated holograms,” J. Mod. Opt. **45**(6), 1231–1237 (1998). [CrossRef]

5. C. Rotschild, S. Zommer, S. Moed, O. Hershcovitz, and S. G. Lipson, “Adjustable spiral phase plate,” Appl. Opt. **43**(12), 2397–2399 (2004). [CrossRef] [PubMed]

6. E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. **94**(23), 231124 (2009). [CrossRef]

7. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. **96**(16), 163905 (2006). [CrossRef] [PubMed]

8. T. A. King, W. Hogervorst, N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Commun. **187**(4-6), 407–414 (2001). [CrossRef]

9. C. F. Phelan, D. P. O’Dwyer, Y. P. Rakovich, J. F. Donegan, and J. G. Lunney, “Conical diffraction and Bessel beam formation with a high optical quality biaxial crystal,” Opt. Express **17**(15), 12891–12899 (2009). [CrossRef] [PubMed]

3. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. **46**(1), 15–28 (2005). [CrossRef]

8. T. A. King, W. Hogervorst, N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Commun. **187**(4-6), 407–414 (2001). [CrossRef]

9. C. F. Phelan, D. P. O’Dwyer, Y. P. Rakovich, J. F. Donegan, and J. G. Lunney, “Conical diffraction and Bessel beam formation with a high optical quality biaxial crystal,” Opt. Express **17**(15), 12891–12899 (2009). [CrossRef] [PubMed]

10. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. **177**(1-6), 297–301 (2000). [CrossRef]

11. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A **4**(4), 651–654 (1987). [CrossRef]

*J*

_{orb}=

*ℓ*ħ, where

*ℓ*is the total topological charge. Thus since the topological charge is necessarily an integer, the OAM of such a beam is quantized. However, in more general cases the OAM is not necessarily connected to the topological charge of the vortices [12

12. M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. **11**(9), 094001 (2009). [CrossRef]

13. J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express **16**(2), 993–1006 (2008). [CrossRef] [PubMed]

14. C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express **14**(15), 6604–6612 (2006). [CrossRef] [PubMed]

5. C. Rotschild, S. Zommer, S. Moed, O. Hershcovitz, and S. G. Lipson, “Adjustable spiral phase plate,” Appl. Opt. **43**(12), 2397–2399 (2004). [CrossRef] [PubMed]

6. E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. **94**(23), 231124 (2009). [CrossRef]

14. C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express **14**(15), 6604–6612 (2006). [CrossRef] [PubMed]

15. S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. **95**(24), 240501 (2005). [CrossRef] [PubMed]

16. A. Vaziri, G. Weihs, and A. Zeilinger, “Superpositions of the orbital angular momentum for applications in quantum experiments,” J. Opt. B Quantum Semiclassical Opt. **4**(2), 367 (2002). [CrossRef]

## 2. Conical diffraction of a Gaussian beam

_{1}< n

_{2}< n

_{3}), such that the wave vector lies along one of the two optic axes then the beam spreads out as a cone in the crystal and emerges as a double-ringed cylindrical beam. Conical refraction was predicted by Hamilton [17] in 1832 and experimentally observed by Lloyd shortly afterwards [18]. For a light beam of finite size it is necessary to account for the corresponding angular spread of the wave vector, therefore it is necessary to treat conical refraction as a diffraction problem. The wave theory of conical diffraction in the paraxial approximation was first treated by Belskii and Khapalyuk [19]. Berry extended this work by showing that the crystal may be represented as a linear operator that transforms the transverse field of the incident light beam [20

20. M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt. **6**(4), 289–300 (2004). [CrossRef]

21. M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observations and theory,” Proc. R. Soc. Lond. A **462**(2070), 1629–1642 (2006). [CrossRef]

*B*

_{0}) and first order (

*B*

_{1}), whose phase and electric field distribution is directly dependent on the input polarisation (

*e*), and is described by Eq. (1)a): where

_{x}, e_{y}*w*is the beam waist,

*J*

_{0}(

*kPR*) and

*J*

_{1}(

*kPR*) are the Bessel functions of zero and first-order and

*a(P)*is the Fourier transform of the transverse profile of the incident field.

*R*

_{0}

*= AL*is the radius of the conically diffracted beam in the focal image plane, where the crystal anisotropy factor

*L*is the length of the crystal. The individual beam profiles

*B*

_{0}

*(R,R*

_{0}

*,Z)*and

*B*

_{1}

*(R,R*

_{0}

*,Z)*satisfy the free space paraxial wave equation and have been shown to correspond very closely to experiment [9

9. C. F. Phelan, D. P. O’Dwyer, Y. P. Rakovich, J. F. Donegan, and J. G. Lunney, “Conical diffraction and Bessel beam formation with a high optical quality biaxial crystal,” Opt. Express **17**(15), 12891–12899 (2009). [CrossRef] [PubMed]

*B*

_{0}component is polarised in the same sense with

*J*

_{sp}= + 1ħ but zero OAM, while the

*B*

_{1}is right-circularly polarised (

*J*

_{sp}= −1ħ) and

*J*

_{orb}= + 1ħ. Thus taken together, the combined beam has zero SAM, but OAM of + ½ħ per photon. This transformation of SAM to OAM in conical diffraction is discussed in more detail by Berry [22

22. M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. **7**(11), 685–690 (2005). [CrossRef]

*B*

_{0}and

*B*

_{1}fields have opposite circular polarisation a circular analyser can be used to isolate

*B*

_{1}, which has OAM of 1ħ per photon and half the optical power of the incident beam.

*B*

_{1}beam with OAM of the same magnitude, also facilitates the generation of non-integer OAM in the range 0 to 1ħ per photon. This can be achieved using an elliptically polarised input beam, which corresponds to non-integer SAM. An elliptically-polarised beam is produced when a quarter-wave plate is placed in a linearly polarised beam so that the incident polarisation makes an angle

*α*with the fast axis. The SAM per photon,

*J*

_{sp}, is given by Eq. (2):

*α*is changed from 0° to 45° the polarisation of the input beam changes from linear with zero SAM, through elliptical with fractional SAM, to circular with SAM of 1ħ.

## 3. Experiment

_{2})

_{4}, supplied by CROptics©. At 632 nm the refractive indices are: n

_{1}= 2.01169, n

_{2}= 2.042198 and n

_{3}= 2.09510 [23

23. D. Kasprowicz, M. Drozdowski, A. Majchrowski, and E. Michalski, “Spectroscopic properties of KGd(WO_{4})_{2}: (Er, Yb) single crystals studied by Brillouin scattering method,” Opt. Mater. **30**(1), 152–154 (2007). [CrossRef]

*R*

_{0}= 5.9x10

^{−4}m and

*B*

_{1}beam a λ/4 plate (P2) and linear polariser were placed after the crystal. The fast axis of wave plate P2 was set orthogonal to the fast axis of P1 and the output linear polariser (LP) was rotated so that it is always orthogonal to the linear polarisation incident on P1. If

*α*is the angle of the input linear polarisation relative to the fast axis of P1, then the polarisation of the light incident on the crystal changes from linear for

*α*= 0°, through elliptical, to left-circular for

*α*= 45°. Using matrix algebra to describe the operations of the various elements of the setup and Eq. (1)a) for the operation of the crystal, it can be shown that for a given angle

*α*, the output optical field is given by:

*B*

_{1}component of the field generated by the crystal. In the case of

*α*= 45° the polarisation into the crystal is left-circular. The square bracket in Eq. (3) reduces to

*α*= 0° the light into the crystal is polarised horizontally and the square bracket in Eq. (3) reduces to -

*i*sin

*θ*. As before, the output is linearly polarised, but now the OAM is zero and there is a sin

^{2}

*θ*azimuthal variation of intensity, with the zero intensity along the horizontal plane, which corresponds to

*θ*= 0°.

**17**(15), 12891–12899 (2009). [CrossRef] [PubMed]

21. M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observations and theory,” Proc. R. Soc. Lond. A **462**(2070), 1629–1642 (2006). [CrossRef]

## 4. Results and discussion

*α*= 45°, i.e. left-circular light into the crystal. As expected the fringe pattern is a spiral, indicative of an optical field with

*J*

_{orb}= + 1ħ per photon. The output and reference beams were then slightly misaligned to generate a wedge fringe pattern. Figure 2(ii-iii) shows the wedge fringe patterns for

*α*= 0° and 45° and Fig. 2(iv-v) shows a Mathematica simulation of the same interference patterns. As expected, the wedge fringe pattern Fig. 2(iii) for circular polarisation input

*α*= 45° shows the expected single fringe dislocation indicative of a beam with

*ℓ*= 1 [2

2. S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. **2**(4), 299–313 (2008). [CrossRef]

*α*was changed from 0° to 45° were also measured. These are shown in Fig. 3(a) , while Fig. 3(b) shows the same intensity distributions calculated using Mathematica using Eq. (3). Linear polarisation into the crystal

*α*= 0° generates a 1st order Hermite-Bessel (HB

_{01}) beam Fig. 3(a-i), with zero intensity on the same axis as the direction of the incident polarisation; there is no fork dislocation in the wedge fringe pattern-Fig. 2(ii) indicates that there is no OAM present. As

*α*is increased from 0° to 45° the beam evolves from a Hermite-Bessel (HB

_{01}) distribution to the 1st order Bessel distribution Fig. 3(a-vi).

22. M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. **7**(11), 685–690 (2005). [CrossRef]

*α*(incident SAM

*J*):

_{sp}*B*

_{1}field is linearly polarised, the SAM is zero. For

*α*= (0°, 8° 16°, 24°,32°,40°,45°) the orbital angular momentum expectation valus

*J*

_{orb}= (0, 0.276, 0.827, 0.852, 0.957, 0.994, 0.999, 1) ħ per photon. The relationship between input SAM and the output OAM can be understood by noting (from Eq. (3)) that the power in the transmitted beam varies as (

*sin*

^{2}(2

*α*) + 1)/2. For

*α*= 45° the optical power is split nearly equally between the

*B*

_{0}(blocked) and

*B*

_{1}(transmitted) fields. Thus at

*α*= 0° the power in the transmitted field is one-quarter of the input power, whereas at

*α*= 45° it is one-half.

24. M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. **24**(7), 430–432 (1999). [CrossRef]

25. W. C. Soares, D. P. Caetano, and J. M. Hickmann, “Hermite-Bessel beams and the geometrical representation of nondiffracting beams with orbital angular momentum,” Opt. Express **14**(11), 4577–4582 (2006). [CrossRef] [PubMed]

*α*= 0° the polarisation into the crystal is linear and the output is a Hermite-Bessel beam with zero OAM. However, for

*α*= 45° the polarisation into the crystal is circular, which may be regarded as a superposition of two equal amplitude orthogonal linear states with a π/2 phase difference. These two linear states generate orthogonal Hermite-Bessel beams which together form a conically diffracting first order Bessel beam with OAM.

## 6. Conclusion

## Acknowledgements

## References and links

1. | L. Allen, S. M. Barnett, and M. J. Padgett, |

2. | S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. |

3. | D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. |

4. | J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer-generated holograms,” J. Mod. Opt. |

5. | C. Rotschild, S. Zommer, S. Moed, O. Hershcovitz, and S. G. Lipson, “Adjustable spiral phase plate,” Appl. Opt. |

6. | E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. |

7. | L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. |

8. | T. A. King, W. Hogervorst, N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Commun. |

9. | C. F. Phelan, D. P. O’Dwyer, Y. P. Rakovich, J. F. Donegan, and J. G. Lunney, “Conical diffraction and Bessel beam formation with a high optical quality biaxial crystal,” Opt. Express |

10. | J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. |

11. | J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A |

12. | M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. |

13. | J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express |

14. | C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express |

15. | S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. |

16. | A. Vaziri, G. Weihs, and A. Zeilinger, “Superpositions of the orbital angular momentum for applications in quantum experiments,” J. Opt. B Quantum Semiclassical Opt. |

17. | W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Transactions of the Royal Irish Academy, 1–144 (1837). |

18. | H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals,” Phil. Mag |

19. | A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. |

20. | M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt. |

21. | M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observations and theory,” Proc. R. Soc. Lond. A |

22. | M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. |

23. | D. Kasprowicz, M. Drozdowski, A. Majchrowski, and E. Michalski, “Spectroscopic properties of KGd(WO |

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

25. | W. C. Soares, D. P. Caetano, and J. M. Hickmann, “Hermite-Bessel beams and the geometrical representation of nondiffracting beams with orbital angular momentum,” Opt. Express |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(260.1180) Physical optics : Crystal optics

(260.1440) Physical optics : Birefringence

(350.5030) Other areas of optics : Phase

**ToC Category:**

Physical Optics

**History**

Original Manuscript: May 20, 2010

Revised Manuscript: June 30, 2010

Manuscript Accepted: July 13, 2010

Published: July 21, 2010

**Citation**

D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. G. Lunney, and J. F. Donegan, "Generation of continuously tunable fractional optical orbital angular momentum using internal conical diffraction," Opt. Express **18**, 16480-16485 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-16480

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### References

- L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (Institute of Physics Publishing, 2003).
- S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photonics Rev. 2(4), 299–313 (2008). [CrossRef]
- D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005). [CrossRef]
- J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer-generated holograms,” J. Mod. Opt. 45(6), 1231–1237 (1998). [CrossRef]
- C. Rotschild, S. Zommer, S. Moed, O. Hershcovitz, and S. G. Lipson, “Adjustable spiral phase plate,” Appl. Opt. 43(12), 2397–2399 (2004). [CrossRef] [PubMed]
- E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, “Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates,” Appl. Phys. Lett. 94(23), 231124 (2009). [CrossRef]
- L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006). [CrossRef] [PubMed]
- T. A. King, W. Hogervorst, N. S. Kazak, N. A. Khilo, and A. A. Ryzhevich, “Formation of higher-order Bessel light beams in biaxial crystals,” Opt. Commun. 187(4-6), 407–414 (2001). [CrossRef]
- C. F. Phelan, D. P. O’Dwyer, Y. P. Rakovich, J. F. Donegan, and J. G. Lunney, “Conical diffraction and Bessel beam formation with a high optical quality biaxial crystal,” Opt. Express 17(15), 12891–12899 (2009). [CrossRef] [PubMed]
- J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177(1-6), 297–301 (2000). [CrossRef]
- J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987). [CrossRef]
- M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. 11(9), 094001 (2009). [CrossRef]
- J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16(2), 993–1006 (2008). [CrossRef] [PubMed]
- C. H. J. Schmitz, K. Uhrig, J. P. Spatz, and J. E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express 14(15), 6604–6612 (2006). [CrossRef] [PubMed]
- S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95(24), 240501 (2005). [CrossRef] [PubMed]
- A. Vaziri, G. Weihs, and A. Zeilinger, “Superpositions of the orbital angular momentum for applications in quantum experiments,” J. Opt. B Quantum Semiclassical Opt. 4(2), 367 (2002). [CrossRef]
- W. R. Hamilton, “Third supplement to an essay on the theory of systems of rays,” Transactions of the Royal Irish Academy, 1–144 (1837).
- H. Lloyd, “On the phenomena presented by light in its passage along the axes of biaxial crystals,” Phil. Mag 1, 112–120 and 207–210 (1833).
- A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 312–315 (1978).
- M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt. 6(4), 289–300 (2004). [CrossRef]
- M. V. Berry, M. R. Jeffrey, and J. G. Lunney, “Conical diffraction: observations and theory,” Proc. R. Soc. Lond. A 462(2070), 1629–1642 (2006). [CrossRef]
- M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005). [CrossRef]
- D. Kasprowicz, M. Drozdowski, A. Majchrowski, and E. Michalski, “Spectroscopic properties of KGd(WO4)2: (Er, Yb) single crystals studied by Brillouin scattering method,” Opt. Mater. 30(1), 152–154 (2007). [CrossRef]
- M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24(7), 430–432 (1999). [CrossRef]
- W. C. Soares, D. P. Caetano, and J. M. Hickmann, “Hermite-Bessel beams and the geometrical representation of nondiffracting beams with orbital angular momentum,” Opt. Express 14(11), 4577–4582 (2006). [CrossRef] [PubMed]

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