Abstract

When a left-circularly polarised Gaussian light beam, which has spin angular momentum (SAM) J sp = σħ = 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 = 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

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2009

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]

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

2008

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]

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

2007

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]

2006

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]

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]

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]

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]

2005

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]

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

2004

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]

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]

2002

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]

2001

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]

2000

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]

1999

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

1998

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]

1987

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

1978

A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 312–315 (1978).

’t Hooft, G. W.

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]

Aiello, A.

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]

Allen, L.

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

Arlt, J.

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. 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]

Barnett, S. M.

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]

Belskii, A. M.

A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 312–315 (1978).

Berry, M. V.

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

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]

Caetano, D. P.

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]

Courtial, J.

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

Curtis, J. E.

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]

Dholakia, K.

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

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. 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]

Donegan, J. F.

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]

Drozdowski, M.

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]

Durnin, J.

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

Eliel, E. R.

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]

Flossmann, F.

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]

Franke-Arnold, S.

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]

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

Götte, J. B.

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]

Hershcovitz, O.

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]

Hickmann, J. M.

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]

Hogervorst, W.

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]

Jeffrey, M. R.

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]

Karimi, E.

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]

Kasprowicz, D.

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]

Kazak, N. S.

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]

Khapalyuk, A. P.

A. M. Belskii and A. P. Khapalyuk, “Internal conical refraction of bounded light beams in biaxial crystals,” Opt. Spectrosc. 44, 312–315 (1978).

Khilo, N. A.

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]

King, T. A.

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]

Lipson, S. G.

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]

Lunney, J. G.

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]

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]

Ma, X.

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]

Majchrowski, A.

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]

Mansuripur, M.

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]

Manzo, C.

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]

Marrucci, L.

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]

McGloin, D.

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

Michalski, E.

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]

Moed, S.

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]

Nagali, E.

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]

O’Dwyer, D. P.

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]

O’Holleran, K.

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]

Oemrawsingh, S. S. R.

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]

Padgett, M.

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

Padgett, M. J.

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]

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

Paparo, D.

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]

Phelan, C. F.

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]

Piccirillo, B.

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]

Preece, D.

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]

Rakovich, Y. P.

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]

Rotschild, C.

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Figures (3)

Fig. 1
Fig. 1

Optical setup to generate a light beam with fractional OAM, and also including a Mach-Zehnder interferometer to record the phase distribution. (See text for explanation)

Fig. 2
Fig. 2

(i) Collinear interference of Gaussian beam with output beam for circularly polarised input. Wedge fringe patterns for non-collinear interference of Gaussian beam with output beam for (ii) α = 0° and (iii) α = 45°. (iv-v) Mathematica simulation of (ii) and (iii).

Fig. 3
Fig. 3

Measured (a) and simulated (b) intensity distributions of the output beam as the polarisation of the input to the crystal is changed by setting the angle α of the linear polarisation relative to the fast axis of the phase plate P1to the following values:(i) 0°, (ii)8°, (iii) 16°, (iv) 24°, (v) 32°, (vi)40°and (vii) 45°.(b) Mathematica simulation of the intensity patterns in (a) using Eq. (3).

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

B 1 ( R , R 0 , Z ) = k 0 P s i n ( k P R 0 ) a ( P ) J 1 ( k P R ) e 1 2 i k p 2 Z d P ,
E ( R , Z ) = B 1 ( R , R 0 , Z ) [ cos θ sin 2 α i sin θ ] ( sin α cos α ) .
J = Im d R ( E * θ E + e z E * × E ) d R E * E ,

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