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13 - Measurement of light's orbital angular momentum

Published online by Cambridge University Press:  05 December 2012

M. P. J. Lavery
University of Glasgow
J. Courtial
University of Glasgow
M. J. Padgett
University of Glasgow
David L. Andrews
University of East Anglia
Mohamed Babiker
University of York
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The orbital angular momentum (OAM) carried by light is widely seen as an extremely useful optical characteristic, with applications in many areas of optics. It was Allen et al. [1] who recognised that a helically phased light beam with a phase cross-section of exp(iℓϕ) carries an OAM, with a value of ℓℏ per photon. Such a light beam contains an optical vortex line of ℓ on its axis. One issue that is yet to be completely resolved is the development of a simple and 100% efficient method for the measurement of OAM.

A better known case of optical angular momentum is spin angular momentum (SAM). SAM is associated with the polarisation state of the light; the spin angular momentum in a left and right circularly polarised beam is σℏ=±1, per photon, respectively [2]. The SAM can be easily determined through the use of a polarising beam splitter, where a π/4 waveplate converts circular polarised light into a p- or s-polarised state which is then transmitted or reflected to give one of two outputs, as shown in Fig. 13.1(a).

OAM arises from the amplitude cross-section of the beam and is therefore independent of the spin angular momentum. One key characteristic of beams carrying OAM is that whereas SAM has only two orthogonal states, the OAM is described by an unbounded state space, i.e. ℓ (as in exp(iℓϕ) can take any integer value [3].

Publisher: Cambridge University Press
Print publication year: 2012

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[1] L., Allen, M., Beijersbergen, R., Spreeuw and J., Woerdman, Orbital angular-momentum of light and the transformation of laguerre-gaussian laser modes, Physical ReviewA, 45(11),8185–9, 1992.Google Scholar
[2] R., Beth, Mechanical detection and measurement of the angular momentum of light, Physical Review, 50, 2, 115–25, 1936.Google Scholar
[3] S., Franke-Arnold, L., Allen and M., Padgett, Advances in optical angular momentum, Laser & Photonics Reviews, 2(4), 299–313, 2008.Google Scholar
[4] H., He, M., Friese and N., Heckenberg, Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity, Physical Review Letters, 75(5), 826–9, 1995.Google Scholar
[5] M., Padgett, J., Arlt, N., Simpson and L., Allen, An experiment to observe the intensity and phase structure of Laguerre–Gaussian laser modes, American Journal Of Physics, 64(1), 77–82, 1996.Google Scholar
[6] 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, Physical Review Letters, 105(15), 153601, 2010.Google Scholar
[7] L., Allen and M., Padgett, Optical tweezers and optical spanners with Laguerre–Gaussian modes, Journal of Modern Optics, 43(12), 2485–91, 1996.Google Scholar
[8] J., Poynting, The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light, Proceedings Of The Royal Society Of London SeriesA, 82, 557, 560–7, 1909.Google Scholar
[9] A., Ashkin, J., Dziedzic, J., Bjorkholm and S., Chu, Observation of a single-beam gradient force optical trap for dielectric particles, Optics Letters, 11(5), 288–90, 1986.Google Scholar
[10] M., Friese, J., Enger and H., Rubinsztein-Dunlop, Optical angular-momentum transfer to trapped absorbing particles, Physical ReviewA, 54(2), 1593–6, 1996.Google Scholar
[11] N., Simpson, L., Allen and M., Padgett, Optical tweezers and optical spanners with Laguerre–Gaussian modes, Journal Of Modern Optics, 43(12), 2485–91, 1996.Google Scholar
[12] K., O'Holleran, M., Padgett and M., Dennis, Topology of optical vortex lines formed by the interference of three, four, and five plane waves, Optics Express, 14(7), 3039–44, 2006.Google Scholar
[13] M., Soskin, V., Gorshkov, M., Vasnetsov, J., Malos and N., Heckenberg, Topological charge and angular momentum of light beams carrying optical vortices, Physical ReviewA, 56(5), 4064–75, 1997.Google Scholar
[14] J., Leach, M., Dennis, J., Courtial and M., Padgett, Laser beams: knotted threads of darkness, Nature, 432(7014), 165–5, 2004.Google Scholar
[15] J. M., Hickmann, E. J. S., Fonseca, W. C., Soares and S., Chavez-Cerda, Unveiling a truncated optical lattice associated with a triangular aperture using light's orbital angular momentum, Physical Review Letters, 105(5), 053904, 2010.Google Scholar
[16] G. C. G., Berkhout and M. W., Beijersbergen, Method for probing the orbital angular momentum of optical vortices in electromagnetic waves from astronomical objects, Physical Review Letters, 101(10), 100801, 2008.Google Scholar
[17] Y., Liu, Measuring the orbital angular momentum of elliptical vortex beams by using a slit hexagon aperture, Optics Communications, 284(10–11), 2424–9, 2011.Google Scholar
[18] A., Mourka, J., Baumgartl and C., Shanor, Visualization of the birth of an optical vortex using diffraction from a triangular aperture, Optics Express, 19(7), 5760–71, 2011.Google Scholar
[19] M., Beijersbergen, L., Allen, H., Vanderveen and J., Woerdman, Astigmatic laser mode converters and transfer of orbital angular-momentum, Optics Communications, 96(1–3), 123–32, 1993.Google Scholar
[20] V., Bazhenov, M., Soskin and M., Vasnetsov, Screw dislocations in light wave-fronts, Journal of Modern Optics, 39(5), 985–90, 1992.Google Scholar
[21] N., Heckenberg, R., McDuff, C., Smith and A., White, Generation of optical-phase singularities by computer-generated holograms, Optics Letters, 17(3), 221–3, 1992.Google Scholar
[22] A., Mair, A., Vaziri, G., Weihs and A., Zeilinger, Entanglement of the orbital angular momentum states of photons, Nature, 412(6844), 313–6, 2001.Google Scholar
[23] G., Molina-Terriza, L., Rebane, J. P., Torres, L., Torner and S., Carrasco, Probing canonical geometrical objects by digital spiral imaging, Journal of the European Optical Society–Rapid Publications, 2, 07014, 2007.Google Scholar
[24] C., Paterson, Atmospheric turbulence and orbital angular momentum of single photons for optical communication, Physical Review Letters, 94, 153901, 2005.Google Scholar
[25] G. A., Tyler and R. W., Boyd, Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum, Optics Letters, 34(2), 142–4, 2009.Google Scholar
[26] G., Gibson, J., Courtial, M., Padgett et al., Free-space information transfer using light beams carrying orbital angular momentum, Optics Express, 12(22), 5448–56, 2004.Google Scholar
[27] S., Khonina, V., Kotlyar and R., Skidanov, Gauss–Laguerre modes with different indices in prescribed diffraction orders of a diffractive phase element, Optics Communications, 175, 301–8, 2000.Google Scholar
[28] M., Vasnetsov, V., Pas'ko and M., Soskin, Analysis of orbital angular momentum of a misaligned optical beam, New Journal of Physics, 7(42), 2005.Google Scholar
[29] S., Franke-Arnold, S. M., Barnett, E., Yao, J., Leach, J., Courtial and M., Padgett, Uncertainty principle for angular position and angular momentum, New Journal of Physics, 6, 103, 2004.Google Scholar
[30] J., Courtial and M., Padgett, Performance of a cylindrical lens mode converter for producing Laguerre–Gaussian laser modes, Optics Communications, 159(1–3), 13–18, 1999.Google Scholar
[31] B., Garetz and S., Arnold, Variable frequency-shifting of circularly polarized laser-radiation via a rotating half-wave retardation plate, Optics Communications, 31(1), 1–3, 1979.Google Scholar
[32] G., Nienhuis, Doppler effect induced by rotating lenses, Optics Communications, 132, 8–14, 1996.Google Scholar
[33] J., Courtial, K., Dholakia, D., Robertson and L., Allen, Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum, Physical Review Letters, 80, 15, 1998.Google Scholar
[34] J., Leach, M., Padgett, S., Barnett, S., Franke-Arnold and J., Courtial, Measuring the orbital angular momentum of a single photon, Physical Review Letters, 88(25), 257901, 2002.Google Scholar
[35] M., Born and E., Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 3rd edn. London: Pergamon Press, 1965.
[36] X., Xue, H., Wei and A., Kirk, Beam analysis by fractional Fourier transform, Optics Letters, 26(22), 1746–8, 2001.Google Scholar
[37] A., Dudley, M., Nock, T., Konrad, F. S., Roux and A., Forbes, Amplitude damping of Laguerre–Gaussian modes, Optics Express, 18(22), 22 789–95, 2010.Google Scholar
[38] L.-P., Deng, H., Wang and K., Wang, Quantum CNOT gates with orbital angular momentum and polarization of single-photon quantum logic, Journal of Modern OpticsB, 24(9), 2517–20, 2007.Google Scholar
[39] O., Bryngdahl, Geometrical transformations in optics, Journal of the Optical Society of America, 64(8), 1092–9, 1974.Google Scholar
[40] W., Hossack, A., Darling and A., Dahdour, Coordinate transformations with multiple computer-generated optical-elements, Journal of Modern Optics, 34(9), 1235–50, 1987.Google Scholar
[41] Y., Saito, S., Komatsu and H., Ohzu, Scale and rotation invariant real-time optical correlator using computer generated hologram, Optics Communications, 47(1), 8–11, 1983.Google Scholar
[42] M. P. J., Lavery, G. C. G., Berkhout, J., Courtial and M. J., Padgett, Measurement of the light orbital angular momentum spectrum using an optical geometric transformation, Journal of Optics, 13(6), 064006, 2011.Google Scholar

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