The coherent superposition of a Gaussian beam with an optical vortex can be mathematically described to occupy the complex plane. We provide a simple analogy between the mathematics, in the form of the complex plane, and the visual representation of these two superimposed optical fields. We provide detailed instructions as to how one can experimentally produce, measure, and control these fields with the use of digital holograms encoded on a spatial light modulator.
References
1.
L.
Allen
, M. W.
Beijersbergen
, R. J. C.
Spreeuw
, and J. P.
Woerdman
, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes
,” Phys. Rev. A
45
, 8185
–8189
(1992
).2.
M. W.
Beijersbergen
, L.
Allen
, H. E. L. O.
Van der Veen
, and J. P.
Woerdman
, “Astigmatic laser mode converters and the transfer of orbital angular momentum
,” Opt. Commun.
96
, 123
–132
(1993
).3.
M. W.
Beijersbergen
, R. P. C.
Coerwinkel
, M.
Kristensen
, and J. P.
Woerdman
, “Helical-wave-front laser-beams produced with a spiral phaseplate
,” Opt. Commun.
112
, 321
–327
(1993
).4.
V. Y.
Bazhenov
, M. V.
Vasnetsov
, and M. S.
Soskin
, “Laser beams with screw dislocations in their wavefronts
,” JETP Lett.
52
, 429
–431
(1990
).5.
N. R.
Heckenberg
, R.
McDuff
, C. P.
Smith
, and A. G.
White
, “Generation of optical phase singularities by computer-generated holograms
,” Opt. Lett.
17
, 221
–223
(1992
).6.
M.
Reicherter
, T.
Haist
, E. U.
Wagemann
, and H. J.
Tiziani
, “Optical particle trapping with computer-generated holograms written on a liquid crystal display
,” Opt. Lett.
24
, 608
–610
(1999
).7.
J. E.
Curtis
, B. A.
Koss
, and D. G.
Grier
, “Dynamic holographic optical tweezers
,” Opt. Commun.
207
, 169
–175
(2002
).8.
S.
Ngcobo
, I.
Litvin
, L.
Burger
, and A.
Forbes
, “A digital laser for on-demand laser modes
,” Nat. Commun.
4
, 2289-1
–2289-6
(2013
).9.
M.
Zürch
, C.
Kern
, P.
Hansinger
, A.
Dreischuh
, and C.
Spielmann
, “Strong-field physics with singular light beams
,” Nat. Phys.
8
, 743
–746
(2012
).10.
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
).11.
E.
Higurashi
, H.
Ukita
, H.
Tanaka
, and O.
Ohguchi
, “Optically induced rotation of anisotropic micro-objects fabricated by surface micromaching
,” App. Phys. Lett.
64
, 2209
–2210
(1994
).12.
P.
Galajda
and P.
Ormosa
, “Complex micromachines produced and driven by light
,” App. Phys. Lett.
78
, 249
–251
(2001
).13.
G.
Knöner
, S.
Parkin
, V. L. Y.
Nieminen
, T. A.
Love
, N. R.
Heckenberg
, and H.
Rubinsztein-Dunlop
, “Integrated optomechanical microelements
,” Opt. Express
15
, 5521
–5530
(2007
).14.
K.
Willig
, S.
Rizzoli
, V.
Westphal
, R.
Jahn
, and S.
Hell
, “STED microscopy reveals that synaptotagmin remains clustered after synaptic vesicle exocytosis
,” Nature
440
, 935
–939
(2006
).15.
G.
Gariepy
, J.
Leach
, K.
Kim
, T.
Hammond
, E.
Frumker
, R.
Boyd
, and P.
Corkum
, “Creating high-harmonic beams with controlled orbital angular momentum
,” Phys. Rev Lett.
113
, 153901
(2014
).16.
C.
Hernández-García
, A.
Picón
, J.
San Román
, and L.
Plaja
, “Attosecond extreme ultraviolet vortices from high-order harmonic generation
,” Phys. Rev Lett.
111
, 083602
(2013
).17.
A.
Mair
, A.
Vasiri
, G.
Weihs
, and A.
Zeilinger
, “Entanglement of the orbital angular momentum states of photons
,” Nature
412
, 313
–316
(2001
).18.
A.
Vasiri
, G.
Weihs
, and A.
Zeilinger
, “Superpositions of the orbital angular momentum for applications in quantum experiments
,” J. Opt. B: Quantum Semiclassical Opt.
4
, S47
–S51
(2002
).19.
B.
Jack
, J.
Leach
, H.
Ritsch
, S. M.
Barnett
, M. J.
Padgett
, and S.
Franke-Arnold
, “Precise quantum tomography of photon pairs with entangled orbital angular momentum
,” New J. Phys.
11
, 103024
(2009
).20.
M.
McLaren
, T.
Mhlanga
, M. J.
Padgett
, F. S.
Roux
, and A.
Forbes
, “Self-healing of quantum entanglement after an obstruction
,” Nat. Commun.
5
, 3248-1
–3248-8
(2013
).21.
G.
Gibson
, J.
Courtial
, M.
Vasnetsov
, M. J.
Padgett
, V.
Pasko
, S. M.
Barnett
, and S.
Franke-Arnold
, “Free-space information transfer using light beams carrying orbital angular momentum
,” Opt. Express
12
, 5448
–5456
(2004
).22.
J.
Wang
, J.
Yang
, I. M.
Fazal
, N.
Ahmed
, Y.
Yan
, H.
Huang
, Y.
Ren
, Y.
Yue
, S.
Dolinar
, M.
Tur
, and A. E.
Willner
, “Terabit free-space data transmission employing orbital angular momentum multiplexing
,” Nature Photon.
6
, 488
–496
(2012
).23.
M.
Mazilu
, A.
Mourka
, T.
Vettenburg
, E. M.
Wright
, and K.
Dholakia
, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum
,” Appl. Phys. Lett.
100
, 231115
(2012
).24.
J. M.
Hickmann
, E. J. S.
Fonseca
, W. C.
Soares
, and S.
Chàvez-Cerda
, “Unveiling a truncated optical lattice associated with a triangular aperture using light's orbital angular momentum
,” Phys. Rev. Lett.
105
, 053904
(2010
).25.
A.
Mourka
, J.
Baumgartl
, C.
Shanor
, K.
Dholakia
, and E. M.
Wright
, “Visualization of the birth of an optical vortex using diffraction from a triangular aperture
,” Opt. Express
19
, 5760
–5771
(2011
).26.
J.
Leach
, M. J.
Padgett
, S. M.
Barnett
, S.
Franke-Arnold
, and J.
Courtial
, “Measuring the orbital angular momentum of a single photon
,” Phys. Rev. Lett.
88
, 257901
(2002
).27.
M. P. J.
Lavery
, A.
Dudley
, A.
Forbes
, J.
Courtial
, and M. J.
Padgett
, “Robust interferometer for the routing of light beams carrying orbital angular momentum
,” New J. Phys.
13
, 093014
(2011
).28.
G. C. G.
Berkhout
, M. P. J.
Lavery
, J.
Courtial
, M. W.
Beijersbergen
, and M. J.
Padgett
, “Efficient sorting of orbital angular momentum states of light
,” Phys. Rev. Lett.
105
, 153601
(2010
).29.
M. P. J.
Lavery
, D. J.
Robertson
, G. C. G.
Berkhout
, G. D.
Love
, M. J.
Padgett
, and J.
Courtial
, “Refractive elements for the measurement of the orbital angular momentum of a single photon
,” Opt. Express
20
, 2110
–2115
(2012
).30.
A.
Dudley
, T.
Mhlanga
, M.
Lavery
, A.
McDonald
, F. S.
Roux
, M. J.
Padgett
, and A.
Forbes
, “Efficient sorting of Bessel beams
,” Opt. Express
21
, 165
–171
(2013
).31.
M. P. J.
Lavery
, D. J.
Robertson
, A.
Sponselli
, J.
Courtial
, N. K.
Steinhoff
, G. A.
Tyler
, A.
Wilner
, and M. J.
Padgett
, “Efficient measurement of orbital angular momentum over 50 states
,” New J. Phys.
15
, 013024
(2013
).32.
A.
Dudley
, G.
Milione
, R. R.
Alfano
, and A.
Forbes
, “All-digital wavefront sensing for structured light beams
,” Opt. Express
22
, 14031
–14040
(2014
).33.
D.
Flamm
, D.
Naidoo
, C.
Schulze
, A.
Forbes
, and M.
Duparré
, “Mode analysis with a spatial light modulator as a correlation filter
,” Opt. Lett.
37
, 2478
–2480
(2012
).34.
T.
Kaiser
, D.
Flamm
, S.
Schröter
, and M.
Duparré
, “Complete modal decomposition for optical fibers using cgh-based correlation filters
,” Opt. Express
17
, 9347
–9356
(2009
).35.
I. A.
Litvin
, A.
Dudley
, F. S.
Roux
, and A.
Forbes
, “Azimuthal decomposition with digital holograms
,” Opt. Express
20
, 10996
–11004
(2012
).36.
D.
Huang
, H.
Timmers
, A.
Roberts
, N.
Shivaram
, and A. S.
Sandhu
, “A low-cost spatial light modulator for use in undergraduate and graduate optics labs
,” Am. J. Phys.
80
, 211
–215
(2012
).37.
J.
Davis
and I.
Moreno
, “Generation of laser beams by digital holograms
,” in Laser Beam Propagation
, edited by A.
Forbes
(CRC Press
, New York
, 2014
), p. 175
.38.
A.
Carpentier
, H.
Michinel
, and J.
Saigueiro
, “Making optical vortices with computer-generated holograms
,” Am. J. Phys.
76
, 916
–921
(2008
).39.
B. R.
Boruah
, “Dynamic manipulation of a laser beam using a liquid crystal spatial light modulator
,” Am. J. Phys.
77
, 331
–336
(2009
).40.
G.
Anzolin
, F.
Tamburini
, A.
Bianchini
, and C.
Barbieri
, “Method to measure off-axis displacements based on the analysis of the intensity distribution of a vortex beam
,” Phys. Rev. A
79
, 033845
(2009
).© 2016 American Association of Physics Teachers.
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