Angular momentum of optical vortex arrays

doi:10.1364/OPEX.14.000938

Optics Express

Editor: Michael Duncan
Vol. 14, Iss. 2 — Jan. 23, 2006
pp: 938–949

Angular momentum of optical vortex arrays

Johannes Courtial, Roberta Zambrini, Mark R. Dennis, and Mikhail Vasnetsov »View Author Affiliations

Optics Express, Vol. 14, Issue 2, pp. 938-949 (2006)
http://dx.doi.org/10.1364/OPEX.14.000938

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Abstract

Guided by the aim to construct light fields with spin-like orbital angular momentum (OAM), that is light fields with a uniform and intrinsic OAM density, we investigate the OAM of strictly periodic arrays of optical vortices with rectangular symmetry. We find that the OAM per unit cell depends on the choice of unit cell and can even change sign when the unit cell is translated. This is the case even if the OAM in each unit cell is intrinsic, that is independent of the choice of measurement axis. We show that spin-like OAM can be found only if the OAM per unit cell vanishes. Our results are applicable to the z component of the angular momentum of any x- and y-periodic momentum distribution in the xy plane, and can also be applied to other periodic light beams and arrays of rotating solids or liquids.

OCIS Codes
(260.0260) Physical optics : Physical optics
(350.0350) Other areas of optics : Other areas of optics

ToC Category:
Physical Optics

Citation
Johannes Courtial, Roberta Zambrini, Mark R. Dennis, and Mikhail Vasnetsov, "Angular momentum of optical vortex arrays," Opt. Express 14, 938-949 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-2-938

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References

J. H. Poynting, "The Wave Motion of a Revolving Shaft, and a Suggestion as to the Angular Momentum in a Beam of Circularly Polarised Light," Proc. R. Soc. London, Ser. A 82, 560-567 (1909). [CrossRef]
L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian modes," Phys. Rev. A 45, 8185-8189 (1992). [CrossRef] [PubMed]
A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053,601 (2002). [CrossRef]
M. Berry, "Paraxial beams of spinning light," in Singular Optics, M. S. Soskin, ed., vol. 3487 of Proc. SPIE, pp. 1-5 (SPIE - the International Society for Optical Engineering, Bellingham, Wash., USA, 1998).
S. M. Barnett, "Optical angular-momentum flux," J. Opt. B: Quantum Semiclass. Opt. 4, S7-S16 (2002). [CrossRef]
I. Dana and I. Freund, "Vortex-lattice wave fields," Opt. Commun. 136, 93-113 (1997). [CrossRef]
R. M. Jenkins, J. Banerji, and A. R. Davies, "The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides," J. Opt. A: Pure Appl. Opt. 3, 527-532 (2001). [CrossRef]
A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, "Generation of lattice structures of optical vortices," J. Opt. Soc. Am. B 19, 550-556 (2002). [CrossRef]
F. S. Roux, "Optical vortex density limitation," Opt. Commun. 223, 31-37 (2003). [CrossRef]
R. Zambrini, L. C. Thomson, S. M. Barnett, and M. Padgett, "Momentum paradox in a vortex core," J. Mod. Opt. 52, 1135-1144 (2005). [CrossRef]
K. Ladavac and D. G. Grier, "Microoptomechanical pumps assembled and driven by holographic optical vortex arrays," Optics Express 12, 1144-1149 (2004). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144</a>. [CrossRef] [PubMed]
M. R. Dennis and J. H. Hannay, "Saddle points in the chaotic analytic function and Ginibre characteristic polynomial," J. Phys. A: Math. Gen. 36, 3379-3383 (2003). [CrossRef]
J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, "Observation of Vortex Lattices in Bose-Einstein Condensates," Science 292, 476-479 (2001). [CrossRef] [PubMed]
R. Donnelly, Quantized Vortices in Helium II (Cambridge University Press, Cambridge, 1991).
P. G. Saffman, Vortex Dynamics (Cambridge University Press, Cambridge, England, 1992).
J. Sommeria, "Experimental study of the two-dimensional inverse energy cascade in a square box," J. Fluid Mech. 170, 139-168 (1986). [CrossRef]
D. L. Boiko, G. Guerrero, and E. Kapon, "Polarization Bloch waves in photonic crystals based on vertical cavity surface emitting laser arrays," Optics Express 12, 2597-2602 (2005). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2597">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2597</a>. [CrossRef]
K. Patorski, "The self-imaging phenomenon and its applications," Progr. Opt. XXVII, 3-108 (1989).
L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Gaussian beams with very high orbital angular momentum," Opt. Commun. 144, 210-213 (1997). [CrossRef]

J. Fluid Mech.

J. Sommeria, "Experimental study of the two-dimensional inverse energy cascade in a square box," J. Fluid Mech. 170, 139-168 (1986). [CrossRef]

J. Mod. Opt.

R. Zambrini, L. C. Thomson, S. M. Barnett, and M. Padgett, "Momentum paradox in a vortex core," J. Mod. Opt. 52, 1135-1144 (2005). [CrossRef]

J. Opt. A: Pure Appl. Opt.

R. M. Jenkins, J. Banerji, and A. R. Davies, "The generation of optical vortices and shape preserving vortex arrays in hollow multimode waveguides," J. Opt. A: Pure Appl. Opt. 3, 527-532 (2001). [CrossRef]

J. Opt. B: Quantum Semiclass. Opt.

S. M. Barnett, "Optical angular-momentum flux," J. Opt. B: Quantum Semiclass. Opt. 4, S7-S16 (2002). [CrossRef]

J. Opt. Soc. Am. B

A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, "Generation of lattice structures of optical vortices," J. Opt. Soc. Am. B 19, 550-556 (2002). [CrossRef]

J. Phys. A: Math. Gen.

M. R. Dennis and J. H. Hannay, "Saddle points in the chaotic analytic function and Ginibre characteristic polynomial," J. Phys. A: Math. Gen. 36, 3379-3383 (2003). [CrossRef]

Opt. Commun.

F. S. Roux, "Optical vortex density limitation," Opt. Commun. 223, 31-37 (2003). [CrossRef]
I. Dana and I. Freund, "Vortex-lattice wave fields," Opt. Commun. 136, 93-113 (1997). [CrossRef]
J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Gaussian beams with very high orbital angular momentum," Opt. Commun. 144, 210-213 (1997). [CrossRef]

Optics Express

K. Ladavac and D. G. Grier, "Microoptomechanical pumps assembled and driven by holographic optical vortex arrays," Optics Express 12, 1144-1149 (2004). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1144</a>. [CrossRef] [PubMed]
D. L. Boiko, G. Guerrero, and E. Kapon, "Polarization Bloch waves in photonic crystals based on vertical cavity surface emitting laser arrays," Optics Express 12, 2597-2602 (2005). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2597">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2597</a>. [CrossRef]

Phys. Rev. A

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian modes," Phys. Rev. A 45, 8185-8189 (1992). [CrossRef] [PubMed]

Phys. Rev. Lett.

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053,601 (2002). [CrossRef]

Progr. Opt.

K. Patorski, "The self-imaging phenomenon and its applications," Progr. Opt. XXVII, 3-108 (1989).

R. Soc. London

J. H. Poynting, "The Wave Motion of a Revolving Shaft, and a Suggestion as to the Angular Momentum in a Beam of Circularly Polarised Light," Proc. R. Soc. London, Ser. A 82, 560-567 (1909). [CrossRef]

Science

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, "Observation of Vortex Lattices in Bose-Einstein Condensates," Science 292, 476-479 (2001). [CrossRef] [PubMed]

SPIE

M. Berry, "Paraxial beams of spinning light," in Singular Optics, M. S. Soskin, ed., vol. 3487 of Proc. SPIE, pp. 1-5 (SPIE - the International Society for Optical Engineering, Bellingham, Wash., USA, 1998).

Other

R. Donnelly, Quantized Vortices in Helium II (Cambridge University Press, Cambridge, 1991).
P. G. Saffman, Vortex Dynamics (Cambridge University Press, Cambridge, England, 1992).
L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

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