Abstract

One of the most widely used techniques for measuring the orbital angular momentum (OAM) components of a light beam is to flatten the spiral phase front of a mode, in order to couple it to a single-mode optical fiber (SMOF). This method, however, suffers from an efficiency that depends on the OAM of the initial mode and on the presence of higher-order radial modes. The reason is that once the phase has been flattened, the field retains its ringed intensity pattern and is therefore a nontrivial superposition of purely radial modes, of which only the fundamental one couples to a SMOF. In this paper, we study the efficiency of this technique both theoretically and experimentally. We find that even for low values of the OAM, a large amount of light can fall outside the fundamental mode of the fiber, and we quantify the losses as functions of the waist of the coupling beam of the OAM and radial indices. Our results can be used as a tool to remove the efficiency bias where fair-sampling loopholes are not a concern. However, we hope that our study will encourage the development of better detection methods of the OAM content of a beam of light.

© 2014 Optical Society of America

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  22. T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in Bell experiments using qudits,” Phys. Rev. Lett. 104, 060401 (2010).
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    [CrossRef]
  24. J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301(R) (2003).
    [CrossRef]
  25. H. Di Lorenzo Pires, H. C. B. Florijn, and M. P. van Exter, “Measurement of the spiral spectrum of entangled two-photon states,” Phys. Rev. Lett. 104, 020505 (2010).
    [CrossRef]
  26. L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre-Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
    [CrossRef]
  27. M. Granata, C. Buy, R. Ward, and M. Barsuglia, “Higher-order Laguerre-Gauss mode generation and interferometry for gravitational wave detectors,” Phys. Rev. Lett. 105, 231102 (2010).
    [CrossRef]
  28. S. Slussarenko, E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Universal unitary gate for single-photon spin-orbit four-dimensional states,” Phys. Rev. A 80, 022326 (2009).
    [CrossRef]
  29. G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, “Probing canonical geometrical objects by digital spiral imaging,” J. Eur. Opt. Soc. 2, 07014 (2007).
    [CrossRef]
  30. V. D’Ambrosio, F. Cardano, E. Karimi, E. Nagali, E. Santamato, L. Marrucci, and F. Sciarrino, “Test of mutually unbiased bases for six-dimensional photonic quantum systems,” Sci. Rep. 3, 2726 (2013).
    [CrossRef]
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    [CrossRef]

2013 (7)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

S. Ngcobo, I. Litvin, L. Burger, and A. Forbes, “A digital laser for on-demand laser modes,” Nat. Commun. 4, 2289 (2013).
[CrossRef]

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[CrossRef]

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre-Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
[CrossRef]

V. D’Ambrosio, F. Cardano, E. Karimi, E. Nagali, E. Santamato, L. Marrucci, and F. Sciarrino, “Test of mutually unbiased bases for six-dimensional photonic quantum systems,” Sci. Rep. 3, 2726 (2013).
[CrossRef]

D. Flamm, C. Schulze, D. Naidoo, S. Schröter, A. Forbes, and M. Duparré, “All-digital holographic tool for mode excitation and analysis in optical fibers,” J. Lightwave Technol. 31, 1023–1032 (2013).
[CrossRef]

E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38, 3546–3549 (2013).
[CrossRef]

2012 (1)

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

2011 (1)

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[CrossRef]

2010 (4)

T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in Bell experiments using qudits,” Phys. Rev. Lett. 104, 060401 (2010).
[CrossRef]

H. Di Lorenzo Pires, H. C. B. Florijn, and M. P. van Exter, “Measurement of the spiral spectrum of entangled two-photon states,” Phys. Rev. Lett. 104, 020505 (2010).
[CrossRef]

G. Berkhout, M. Lavery, J. Courtial, M. Beijersbergen, and M. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 105601 (2010).
[CrossRef]

M. Granata, C. Buy, R. Ward, and M. Barsuglia, “Higher-order Laguerre-Gauss mode generation and interferometry for gravitational wave detectors,” Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

2009 (3)

S. Slussarenko, E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Universal unitary gate for single-photon spin-orbit four-dimensional states,” Phys. Rev. A 80, 022326 (2009).
[CrossRef]

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, 231124 (2009).
[CrossRef]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

2007 (5)

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, “Probing canonical geometrical objects by digital spiral imaging,” J. Eur. Opt. Soc. 2, 07014 (2007).
[CrossRef]

W. S. Hell, “Far-field optical nanoscopy,” Science 316, 1153–1158 (2007).
[CrossRef]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

T. Grosjean, D. Courjon, and C. Bainier, “Smallest lithographic marks generated by optical focusing systems,” Opt. Lett. 32, 976–978 (2007).
[CrossRef]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett. 32, 3053–3055 (2007).
[CrossRef]

2006 (1)

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

2005 (1)

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

2004 (1)

2003 (1)

J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301(R) (2003).
[CrossRef]

2001 (2)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[CrossRef]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef]

1995 (1)

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with high order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

1992 (2)

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. 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).
[CrossRef]

Ahmed, N.

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Alexandrescu, A.

J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301(R) (2003).
[CrossRef]

Allen, L.

L. Allen, M. W. Beijersbergen, R. 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).
[CrossRef]

Andersson, E.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[CrossRef]

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef]

Bainier, C.

Barnett, S. M.

Barsuglia, M.

M. Granata, C. Buy, R. Ward, and M. Barsuglia, “Higher-order Laguerre-Gauss mode generation and interferometry for gravitational wave detectors,” Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

Bazhenov, V. Y.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Beijersbergen, M.

G. Berkhout, M. Lavery, J. Courtial, M. Beijersbergen, and M. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 105601 (2010).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. 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).
[CrossRef]

Bent, N.

Berkhout, G.

G. Berkhout, M. Lavery, J. Courtial, M. Beijersbergen, and M. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 105601 (2010).
[CrossRef]

Bogan, C.

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre-Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
[CrossRef]

Bolduc, E.

Boyd, R. W.

E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38, 3546–3549 (2013).
[CrossRef]

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[CrossRef]

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

Brunner, N.

T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in Bell experiments using qudits,” Phys. Rev. Lett. 104, 060401 (2010).
[CrossRef]

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef]

Buller, G. S.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[CrossRef]

Burger, L.

S. Ngcobo, I. Litvin, L. Burger, and A. Forbes, “A digital laser for on-demand laser modes,” Nat. Commun. 4, 2289 (2013).
[CrossRef]

Buy, C.

M. Granata, C. Buy, R. Ward, and M. Barsuglia, “Higher-order Laguerre-Gauss mode generation and interferometry for gravitational wave detectors,” Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

Carbone, L.

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre-Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
[CrossRef]

Cardano, F.

V. D’Ambrosio, F. Cardano, E. Karimi, E. Nagali, E. Santamato, L. Marrucci, and F. Sciarrino, “Test of mutually unbiased bases for six-dimensional photonic quantum systems,” Sci. Rep. 3, 2726 (2013).
[CrossRef]

Carrasco, S.

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, “Probing canonical geometrical objects by digital spiral imaging,” J. Eur. Opt. Soc. 2, 07014 (2007).
[CrossRef]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Courjon, D.

Courtial, J.

G. Berkhout, M. Lavery, J. Courtial, M. Beijersbergen, and M. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 105601 (2010).
[CrossRef]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, 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).
[CrossRef]

D’Ambrosio, V.

V. D’Ambrosio, F. Cardano, E. Karimi, E. Nagali, E. Santamato, L. Marrucci, and F. Sciarrino, “Test of mutually unbiased bases for six-dimensional photonic quantum systems,” Sci. Rep. 3, 2726 (2013).
[CrossRef]

Dada, A. C.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[CrossRef]

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Dholakia, K.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef]

Di Lorenzo Pires, H.

H. Di Lorenzo Pires, H. C. B. Florijn, and M. P. van Exter, “Measurement of the spiral spectrum of entangled two-photon states,” Phys. Rev. Lett. 104, 020505 (2010).
[CrossRef]

Dolinar, S.

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Duparré, M.

Fazal, I. M.

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Flamm, D.

Florijn, H. C. B.

H. Di Lorenzo Pires, H. C. B. Florijn, and M. P. van Exter, “Measurement of the spiral spectrum of entangled two-photon states,” Phys. Rev. Lett. 104, 020505 (2010).
[CrossRef]

Forbes, A.

Franke-Arnold, S.

Freise, A.

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre-Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
[CrossRef]

Fulda, P.

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre-Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
[CrossRef]

Gibson, G.

Granata, M.

M. Granata, C. Buy, R. Ward, and M. Barsuglia, “Higher-order Laguerre-Gauss mode generation and interferometry for gravitational wave detectors,” Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

Grosjean, T.

He, H.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with high order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

Heckenberg, N. R.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with high order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

Hell, W. S.

W. S. Hell, “Far-field optical nanoscopy,” Science 316, 1153–1158 (2007).
[CrossRef]

Huang, H.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Karimi, E.

E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38, 3546–3549 (2013).
[CrossRef]

V. D’Ambrosio, F. Cardano, E. Karimi, E. Nagali, E. Santamato, L. Marrucci, and F. Sciarrino, “Test of mutually unbiased bases for six-dimensional photonic quantum systems,” Sci. Rep. 3, 2726 (2013).
[CrossRef]

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, 231124 (2009).
[CrossRef]

S. Slussarenko, E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Universal unitary gate for single-photon spin-orbit four-dimensional states,” Phys. Rev. A 80, 022326 (2009).
[CrossRef]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett. 32, 3053–3055 (2007).
[CrossRef]

Kristensen, M.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Kristensen, P.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

Lavery, M.

G. Berkhout, M. Lavery, J. Courtial, M. Beijersbergen, and M. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 105601 (2010).
[CrossRef]

Leach, J.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[CrossRef]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Litvin, I.

S. Ngcobo, I. Litvin, L. Burger, and A. Forbes, “A digital laser for on-demand laser modes,” Nat. Commun. 4, 2289 (2013).
[CrossRef]

MacDonald, M. P.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[CrossRef]

Malik, M.

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[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, 163905 (2006).
[CrossRef]

Marrucci, L.

V. D’Ambrosio, F. Cardano, E. Karimi, E. Nagali, E. Santamato, L. Marrucci, and F. Sciarrino, “Test of mutually unbiased bases for six-dimensional photonic quantum systems,” Sci. Rep. 3, 2726 (2013).
[CrossRef]

S. Slussarenko, E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Universal unitary gate for single-photon spin-orbit four-dimensional states,” Phys. Rev. A 80, 022326 (2009).
[CrossRef]

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, 231124 (2009).
[CrossRef]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett. 32, 3053–3055 (2007).
[CrossRef]

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

Mirhosseini, M.

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[CrossRef]

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, “Probing canonical geometrical objects by digital spiral imaging,” J. Eur. Opt. Soc. 2, 07014 (2007).
[CrossRef]

Nagali, E.

V. D’Ambrosio, F. Cardano, E. Karimi, E. Nagali, E. Santamato, L. Marrucci, and F. Sciarrino, “Test of mutually unbiased bases for six-dimensional photonic quantum systems,” Sci. Rep. 3, 2726 (2013).
[CrossRef]

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, 231124 (2009).
[CrossRef]

Naidoo, D.

Ngcobo, S.

S. Ngcobo, I. Litvin, L. Burger, and A. Forbes, “A digital laser for on-demand laser modes,” Nat. Commun. 4, 2289 (2013).
[CrossRef]

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

Padgett, M.

G. Berkhout, M. Lavery, J. Courtial, M. Beijersbergen, and M. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 105601 (2010).
[CrossRef]

Padgett, M. J.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[CrossRef]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, 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).
[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, 163905 (2006).
[CrossRef]

Pasko, V.

Paterson, L.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef]

Piccirillo, B.

S. Slussarenko, E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Universal unitary gate for single-photon spin-orbit four-dimensional states,” Phys. Rev. A 80, 022326 (2009).
[CrossRef]

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, 231124 (2009).
[CrossRef]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett. 32, 3053–3055 (2007).
[CrossRef]

Pironio, S.

T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in Bell experiments using qudits,” Phys. Rev. Lett. 104, 060401 (2010).
[CrossRef]

Ramachandran, S.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

Rebane, L.

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, “Probing canonical geometrical objects by digital spiral imaging,” J. Eur. Opt. Soc. 2, 07014 (2007).
[CrossRef]

Ren, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Rubinsztein-Dunlop, H.

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with high order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

Santamato, E.

E. Bolduc, N. Bent, E. Santamato, E. Karimi, and R. W. Boyd, “Exact solution to simultaneous intensity and phase encryption with a single phase-only hologram,” Opt. Lett. 38, 3546–3549 (2013).
[CrossRef]

V. D’Ambrosio, F. Cardano, E. Karimi, E. Nagali, E. Santamato, L. Marrucci, and F. Sciarrino, “Test of mutually unbiased bases for six-dimensional photonic quantum systems,” Sci. Rep. 3, 2726 (2013).
[CrossRef]

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, 231124 (2009).
[CrossRef]

S. Slussarenko, E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Universal unitary gate for single-photon spin-orbit four-dimensional states,” Phys. Rev. A 80, 022326 (2009).
[CrossRef]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian modes,” Opt. Lett. 32, 3053–3055 (2007).
[CrossRef]

Schröter, S.

Schulze, C.

Sciarrino, F.

V. D’Ambrosio, F. Cardano, E. Karimi, E. Nagali, E. Santamato, L. Marrucci, and F. Sciarrino, “Test of mutually unbiased bases for six-dimensional photonic quantum systems,” Sci. Rep. 3, 2726 (2013).
[CrossRef]

Shi, Z.

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[CrossRef]

Sibbett, W.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef]

Slussarenko, S.

S. Slussarenko, E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Universal unitary gate for single-photon spin-orbit four-dimensional states,” Phys. Rev. A 80, 022326 (2009).
[CrossRef]

Soskin, M. S.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Spreeuw, R.

L. Allen, M. W. Beijersbergen, R. 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).
[CrossRef]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, “Probing canonical geometrical objects by digital spiral imaging,” J. Eur. Opt. Soc. 2, 07014 (2007).
[CrossRef]

J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301(R) (2003).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, “Probing canonical geometrical objects by digital spiral imaging,” J. Eur. Opt. Soc. 2, 07014 (2007).
[CrossRef]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301(R) (2003).
[CrossRef]

Tur, M.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

van Exter, M. P.

H. Di Lorenzo Pires, H. C. B. Florijn, and M. P. van Exter, “Measurement of the spiral spectrum of entangled two-photon states,” Phys. Rev. Lett. 104, 020505 (2010).
[CrossRef]

Vasnetsov, M.

Vasnetsov, M. V.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[CrossRef]

Vértesi, T.

T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in Bell experiments using qudits,” Phys. Rev. Lett. 104, 060401 (2010).
[CrossRef]

Wang, J.

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Ward, R.

M. Granata, C. Buy, R. Ward, and M. Barsuglia, “Higher-order Laguerre-Gauss mode generation and interferometry for gravitational wave detectors,” Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[CrossRef]

Willke, B.

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre-Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
[CrossRef]

Willner, A. E.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Woerdman, J. P.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. 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).
[CrossRef]

Yan, Y.

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Yang, J.-Y.

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Yue, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[CrossRef]

Zito, G.

Appl. Phys. Lett. (1)

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, 231124 (2009).
[CrossRef]

J. Eur. Opt. Soc. (1)

G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner, and S. Carrasco, “Probing canonical geometrical objects by digital spiral imaging,” J. Eur. Opt. Soc. 2, 07014 (2007).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (2)

H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with high order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. 42, 217–223 (1995).
[CrossRef]

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Nat. Commun. (2)

S. Ngcobo, I. Litvin, L. Burger, and A. Forbes, “A digital laser for on-demand laser modes,” Nat. Commun. 4, 2289 (2013).
[CrossRef]

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[CrossRef]

Nat. Photonics (1)

J. Wang, J.-Y. 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,” Nat. Photonics 6, 488–496 (2012).
[CrossRef]

Nat. Phys. (2)

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[CrossRef]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[CrossRef]

Nature (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
[CrossRef]

New J. Phys. (1)

J. Leach, M. R. Dennis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Opt. Commun. (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (3)

S. Slussarenko, E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, “Universal unitary gate for single-photon spin-orbit four-dimensional states,” Phys. Rev. A 80, 022326 (2009).
[CrossRef]

J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301(R) (2003).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. 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).
[CrossRef]

Phys. Rev. Lett. (6)

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

G. Berkhout, M. Lavery, J. Courtial, M. Beijersbergen, and M. Padgett, “Efficient sorting of orbital angular momentum states of light,” Phys. Rev. Lett. 105, 105601 (2010).
[CrossRef]

H. Di Lorenzo Pires, H. C. B. Florijn, and M. P. van Exter, “Measurement of the spiral spectrum of entangled two-photon states,” Phys. Rev. Lett. 104, 020505 (2010).
[CrossRef]

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre-Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
[CrossRef]

M. Granata, C. Buy, R. Ward, and M. Barsuglia, “Higher-order Laguerre-Gauss mode generation and interferometry for gravitational wave detectors,” Phys. Rev. Lett. 105, 231102 (2010).
[CrossRef]

T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in Bell experiments using qudits,” Phys. Rev. Lett. 104, 060401 (2010).
[CrossRef]

Prog. Opt. (1)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

Sci. Rep. (1)

V. D’Ambrosio, F. Cardano, E. Karimi, E. Nagali, E. Santamato, L. Marrucci, and F. Sciarrino, “Test of mutually unbiased bases for six-dimensional photonic quantum systems,” Sci. Rep. 3, 2726 (2013).
[CrossRef]

Science (3)

W. S. Hell, “Far-field optical nanoscopy,” Science 316, 1153–1158 (2007).
[CrossRef]

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2001).
[CrossRef]

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Intensity of the p=0 (left) and p=1 (right) modes at the input to the fiber. As an effect of diffraction, local maxima at the periphery gain intensity as p and || increase. Here ρ is in units of a0=(2λf)/(πw0), which is the natural scaling factor in the far field of the lens.

Fig. 2.
Fig. 2.

Coupling efficiency for projective measurements for the modes in Fig. 1. For a given choice of optics, the coupling efficiency shows a bias dependent on the order of the transverse modes. The horizontal axis is scanned by changing w0, as a0 is inversely proportional to w0. The shaded box indicates the region limited by the active area of the SLMs (see experimental section).

Fig. 3.
Fig. 3.

Experimental setup for generating and detecting photon transverse states. A linearly polarized He–Ne laser beam is spatially cleaned with two lenses and a pinhole. A half-wave plate (HWP) optimizes the first order of diffraction on SLMA, since SLMs are polarization dependent. The mode LGp,(r) produced by SLMA is then projected on the mode LGp,(r)* on SLMB. The resulting far field is coupled into a SMOF. We implement two 4f systems with unit magnification and a microscope objective to image SLMA on SLMB and SLMB on the microscope objective. Irises are used to select the first order of diffraction at the far-field plane of SLMs, where higher orders of diffraction are well separated.

Fig. 4.
Fig. 4.

Experimentally measured overall coupling efficiency for the modes shown in Fig. 1: (left) =05 and (right) =02. The shaded regions indicate a domain in which the effective beam size exceeds the active area of the SLM, resulting in unreliable data.

Equations (8)

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

LGp,(r,ϕ)2||+1p!πw02(p+||)!(rw0)||er2w02Lp||(2r2w02)eiϕ,
Fp,(ρ,φ)=FT[LGp,(r)LGp,*(r)],
Fp,(ρ,φ)=2πeiπλfρ2iλf0rdr|LGp,(r)|2J0(2πλfrρ).
η=2πσ2|0ρdρ02πdφF(ρ,φ)eρ2σ2|2,
η0=||!2(2||)!A2||+1B,
η1=(||+1)!||!4(2+3||)(2||)!A2||+1B(A2+B2(||+1))2,
ηp=|Npj=0p(1)j(pj)ajfjp,|2
fjp,=k=1p(||+k+kH(jk)),aj=2πσ/a0(1+(σ/a0)2)||2+j+1,

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