Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group

Modal analysis of structured light with spatial light modulators: a practical tutorial

Open Access Open Access

Abstract

A quantitative analysis of optical fields is essential, particularly when the light is structured in some desired manner, or when there is perhaps an undesired structure that must be corrected for. A ubiquitous procedure in the optical community is that of optical mode projections—a modal analysis of light—for the unveiling of amplitude and phase information of a light field. When correctly performed, all the salient features of the field can be deduced with high fidelity, including its orbital angular momentum, vectorial properties, wavefront, and Poynting vector. Here, we present a practical tutorial on how to perform an efficient and effective optical modal decomposition, with emphasis on holographic approaches using spatial light modulators, highlighting the care required at each step of the process.

© 2020 Optical Society of America

Full Article  |  PDF Article
More Like This
Modal description of paraxial structured light propagation: tutorial

Hend Sroor, Chane Moodley, Valeria Rodríguez-Fajardo, Qiwen Zhan, and Andrew Forbes
J. Opt. Soc. Am. A 38(10) 1443-1449 (2021)

Digital Stokes polarimetry and its application to structured light: tutorial

Keshaan Singh, Najmeh Tabebordbar, Andrew Forbes, and Angela Dudley
J. Opt. Soc. Am. A 37(11) C33-C44 (2020)

High-dimensional cryptography with spatial modes of light: tutorial

Eileen Otte, Isaac Nape, Carmelo Rosales-Guzmán, Cornelia Denz, Andrew Forbes, and Bienvenu Ndagano
J. Opt. Soc. Am. B 37(11) A309-A323 (2020)

References

  • View by:

  1. B. R. Boruah, “Dynamic manipulation of a laser beam using a liquid crystal spatial light modulator,” Am. J. Phys. 77, 331–336 (2009).
    [Crossref]
  2. A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photon. 8, 200–227 (2016).
    [Crossref]
  3. G. Lazarev, P.-J. Chen, J. Strauss, N. Fontaine, and A. Forbes, “Beyond the display: phase-only liquid crystal on silicon devices and their applications in photonics,” Opt. Express 27, 16206–16249 (2019).
    [Crossref]
  4. Y. X. Ren, M. Li, K. Huang, J. G. Wu, H. F. Gao, Z. Q. Wang, and Y. M. Li, “Experimental generation of Laguerre-Gaussian beam using digital micromirror device,” Appl. Opt. 49, 1838–1844 (2010).
    [Crossref]
  5. M. Mirhosseini, O. S. Magaña-Loaiza, C. Chen, B. Rodenburg, M. Malik, and R. W. Boyd, “Rapid generation of light beams carrying orbital angular momentum,” Opt. Express 21, 30196–30203 (2013).
    [Crossref]
  6. M. A. Cox, E. Toninelli, L. Cheng, M. J. Padgett, and A. Forbes, “A high-speed, wavelength invariant, single-pixel wavefront sensor with a digital micromirror device,” IEEE Access 7, 85860–85866 (2019).
    [Crossref]
  7. S. Scholes, R. Kara, J. Pinnell, V. Rodríguez-Fajardo, and A. Forbes, “Structured light with digital micromirror devices: a guide to best practice,” Opt. Eng. 59, 041202 (2019).
    [Crossref]
  8. H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
    [Crossref]
  9. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412, 313–316 (2001).
    [Crossref]
  10. A. Forbes and I. Nape, “Quantum mechanics with patterns of light: progress in high dimensional and multidimensional entanglement with structured light,” AVS Quantum Sci. 1, 011701 (2019).
    [Crossref]
  11. A. E. Siegman, Defining and Measuring Laser Beam Quality (Springer, 1993), pp. 13–28.
  12. A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
    [Crossref]
  13. M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
    [Crossref]
  14. B. Ndagano, H. Sroor, M. McLaren, C. Rosales-Guzmán, and A. Forbes, “Beam quality measure for vector beams,” Opt. Lett. 41, 3407–3410 (2016).
    [Crossref]
  15. 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).
    [Crossref]
  16. N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10, 1865 (2019).
    [Crossref]
  17. Y. Zhou, J. Zhao, Z. Shi, S. M. H. Rafsanjani, M. Mirhosseini, Z. Zhu, A. E. Willner, and R. W. Boyd, “Hermite–Gaussian mode sorter,” Opt. Lett. 43, 5263–5266 (2018).
    [Crossref]
  18. A. Dudley, T. Mhlanga, M. Lavery, A. McDonald, F. S. Roux, M. Padgett, and A. Forbes, “Efficient sorting of Bessel beams,” Opt. Express 21, 165–171 (2013).
    [Crossref]
  19. C. Rosales-Guzmán, N. Bhebhe, and A. Forbes, “Simultaneous generation of multiple vector beams on a single SLM,” Opt. Express 25, 25697–25706 (2017).
    [Crossref]
  20. B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Lightwave Technol. 36, 292–301 (2018).
    [Crossref]
  21. B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. J. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep. 7, 13882 (2017).
    [Crossref]
  22. G. Ruffato, M. Girardi, M. Massari, E. Mafakheri, B. Sephton, P. Capaldo, A. Forbes, and F. Romanato, “A compact diffractive sorter for high-resolution demultiplexing of orbital angular momentum beams,” Sci. Rep. 8, 10248 (2018).
    [Crossref]
  23. G. Ruffato, M. Massari, M. Girardi, G. Parisi, M. Zontini, and F. Romanato, “Non-paraxial design and fabrication of a compact OAM sorter in the telecom infrared,” Opt. Express 27, 24123–24134 (2019).
    [Crossref]
  24. S. Pachava, R. Dharmavarapu, A. Vijayakumar, S. Jayakumar, A. Manthalkar, A. Dixit, N. K. Viswanathan, B. Srinivasan, and S. Bhattacharya, “Generation and decomposition of scalar and vector modes carrying orbital angular momentum: a review,” Opt. Eng. 59, 041205 (2019).
    [Crossref]
  25. M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
    [Crossref]
  26. Z. Shi, M. Mirhosseini, J. Margiewicz, M. Malik, F. Rivera, Z. Zhu, and R. W. Boyd, “Scan-free direct measurement of an extremely high-dimensional photonic state,” Optica 2, 388–392 (2015).
    [Crossref]
  27. D. Mardani, A. F. Abouraddy, and G. K. Atia, “Interferometry-based modal analysis with finite aperture effects,” J. Opt. Soc. Am. A 35, 1880–1890 (2018).
    [Crossref]
  28. 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).
    [Crossref]
  29. 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).
    [Crossref]
  30. G. Kulkarni, R. Sahu, O. S. Magaña-Loaiza, R. W. Boyd, and A. K. Jha, “Single-shot measurement of the orbital-angular-momentum spectrum of light,” Nat. Commun. 8, 1054 (2017).
    [Crossref]
  31. J. Hickmann, E. Fonseca, W. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
    [Crossref]
  32. 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).
    [Crossref]
  33. 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).
    [Crossref]
  34. M. A. Golub, I. N. Sisakian, V. A. Soifer, and G. V. Uvarov, “New measurement techniques for modal power distribution in fibers,” Proc. SPIE 1366, 273–282 (1991).
    [Crossref]
  35. O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94, 143902 (2005).
    [Crossref]
  36. D. Flamm, C. Schulze, D. Naidoo, S. Schroter, 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]
  37. L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
    [Crossref]
  38. A. Trichili, K.-H. Park, M. Zghal, B. S. Ooi, and M.-S. Alouini, “Communicating using spatial mode multiplexing: potentials, challenges and perspectives,” Commun. Surveys Tuts. 21, 3175–3203 (2019).
    [Crossref]
  39. A. E. Siegman, “Basic concepts,” in Lasers (University Science Books, 1986), Chap. 2, pp. 7–15.
  40. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [Crossref]
  41. A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [Crossref]
  42. M. A. Bandres and J. C. Gutiérrez-Vega, “Ince–Gaussian modes of the paraxial wave equation and stable resonators,” J. Opt. Soc. Am. A 21, 873–880 (2004).
    [Crossref]
  43. M. A. Bandres and J. C. Gutiérrez-Vega, “Ince–Gaussian beams,” Opt. Lett. 29, 144–146 (2004).
    [Crossref]
  44. J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid-crystal display,” Opt. Lett. 31, 649–651 (2006).
    [Crossref]
  45. J. P. Kirk and A. L. Jones, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am. 61, 1023–1028 (1971).
    [Crossref]
  46. M. R. Duparré, V. S. Pavelyev, B. Luedge, E.-B. Kley, V. A. Soifer, and R. M. Kowarschik, “Generation, superposition, and separation of Gauss-Hermite modes by means of does,” Proc. SPIE 3291, 104–114 (1998).
    [Crossref]
  47. M. R. Duparré, C. Rockstuhl, A. Letsch, S. Schroeter, and V. S. Pavelyev, “On-line control of laser beam quality by means of diffractive optical components,” Proc. SPIE 4932, 549–559 (2003).
    [Crossref]
  48. 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).
    [Crossref]
  49. I. A. Litvin, A. Dudley, F. S. Roux, and A. Forbes, “Azimuthal decomposition with digital holograms,” Opt. Express 20, 10996–11004 (2012).
    [Crossref]
  50. 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).
    [Crossref]
  51. A. Trichili, T. Mhlanga, Y. Ismail, F. S. Roux, M. McLaren, M. Zghal, and A. Forbes, “Detection of Bessel beams with digital axicons,” Opt. Express 22, 17553–17560 (2014).
    [Crossref]
  52. C. Rosales-Guzmán and A. Forbes, How to Shape Light with Spatial Light Modulators (SPIE, 2017).
  53. K. Choi and C. Jun, “High-precision modal decomposition of laser beams based on globally optimized SPGD algorithm,” IEEE Photon. J. 11, 7103110 (2019).
    [Crossref]
  54. R. Lindzen, “A modal decomposition of the semidiurnal tide in the lower thermosphere,” J. Geophys. Res. 81, 2923–2926 (1976).
    [Crossref]
  55. T. Yang, “A method of range and depth estimation by modal decomposition,” J. Acoust. Soc. Am. 82, 1736–1745 (1987).
    [Crossref]
  56. E. J. R. Vesseur, F. J. García de Abajo, and A. Polman, “Modal decomposition of surface–plasmon whispering gallery resonators,” Nano Lett. 9, 3147–3150 (2009).
    [Crossref]
  57. G. Boudarham and M. Kociak, “Modal decompositions of the local electromagnetic density of states and spatially resolved electron energy loss probability in terms of geometric modes,” Phys. Rev. B 85, 245447 (2012).
    [Crossref]
  58. F. Bouchard, N. H. Valencia, F. Brandt, R. Fickler, M. Huber, and M. Malik, “Measuring azimuthal and radial modes of photons,” Opt. Express 26, 31925–31941 (2018).
    [Crossref]
  59. H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6, e16251 (2017).
    [Crossref]
  60. A. D’Errico, R. D’Amelio, B. Piccirillo, F. Cardano, and L. Marrucci, “Measuring the complex orbital angular momentum spectrum and spatial mode decomposition of structured light beams,” Optica 4, 1350–1357 (2017).
    [Crossref]
  61. C. Schulze, D. Flamm, M. Duparré, and A. Forbes, “Beam-quality measurements using a spatial light modulator,” Opt. Lett. 37, 4687–4689 (2012).
    [Crossref]
  62. C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20, 19714–19725 (2012).
    [Crossref]
  63. D. Flamm, C. Schulze, R. Brüning, O. A. Schmidt, T. Kaiser, S. Schröter, and M. Duparré, “Fast M2 measurement for fiber beams based on modal analysis,” Appl. Opt. 51, 987–993 (2012).
    [Crossref]
  64. C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
    [Crossref]
  65. C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Reconstruction of laser beam wavefronts based on mode analysis,” Appl. Opt. 52, 5312–5317 (2013).
    [Crossref]
  66. D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010).
    [Crossref]
  67. A. Dudley, I. Litvin, and A. Forbes, “Quantitative measurement of the orbital angular momentum density of light,” Appl. Opt. 51, 823–833 (2012).
    [Crossref]
  68. G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801 (2009).
    [Crossref]
  69. L. Changhai, X. Fengjie, H. Shengyang, and J. Zongfu, “Performance analysis of multiplexed phase computer-generated hologram for modal wavefront sensing,” Appl. Opt. 50, 1631–1639 (2011).
    [Crossref]
  70. M. R. Duparré, V. S. Pavelyev, V. A. Soifer, and B. Luedge, “Laser beam characterization by means of diffractive optical correlation filters,” Proc. SPIE 4095, 40–48 (2000).
    [Crossref]
  71. B. R. Boruah and M. A. Neil, “Susceptibility to and correction of azimuthal aberrations in singular light beams,” Opt. Express 14, 10377–10385 (2006).
    [Crossref]
  72. O. A. Schmidt, C. Schulze, D. Flamm, R. Brüning, T. Kaiser, S. Schröter, and M. Duparré, “Real-time determination of laser beam quality by modal decomposition,” Opt. Express 19, 6741–6748 (2011).
    [Crossref]
  73. F. Stutzki, H. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett. 36, 4572–4574 (2011).
    [Crossref]
  74. M. Duparré, B. Lüdge, and S. Schröter, “On-line characterization of Nd:YAG laser beams by means of modal decomposition using diffractive optical correlation filters,” Proc. SPIE 5962, 755–764 (2005).
    [Crossref]
  75. V. A. Soifer and M. A. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).
  76. D. Naidoo, K. At-Ameur, M. Brunel, and A. Forbes, “Intra-cavity generation of superpositions of Laguerre–Gaussian beams,” Appl. Phys. B 106, 683–690 (2012).
    [Crossref]
  77. D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
    [Crossref]
  78. M. Paurisse, M. Hanna, F. Druon, P. Georges, C. Bellanger, A. Brignon, and J. Huignard, “Phase and amplitude control of a multimode LMA fiber beam by use of digital holography,” Opt. Express 17, 13000–13008 (2009).
    [Crossref]
  79. M. Paurisse, L. Lévèque, M. Hanna, F. Druon, and P. Georges, “Complete measurement of fiber modal content by wavefront analysis,” Opt. Express 20, 4074–4084 (2012).
    [Crossref]
  80. C. Schulze, A. Lorenz, D. Flamm, A. Hartung, S. Schröter, H. Bartelt, and M. Duparré, “Mode resolved bend loss in few-mode optical fibers,” Opt. Express 21, 3170–3181 (2013).
    [Crossref]
  81. V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A 24, 3500–3507 (2007).
    [Crossref]
  82. T. W. Clark, R. F. Offer, S. Franke-Arnold, A. S. Arnold, and N. Radwell, “Comparison of beam generation techniques using a phase only spatial light modulator,” Opt. Express 24, 6249–6264 (2016).
    [Crossref]
  83. F. S. Roux and Y. Zhang, “Projective measurements in quantum and classical optical systems,” Phys. Rev. A 90, 033835 (2014).
    [Crossref]
  84. H. Qassim, F. M. Miatto, J. P. Torres, M. J. Padgett, E. Karimi, and R. W. Boyd, “Limitations to the determination of a Laguerre–Gauss spectrum via projective, phase-flattening measurement,” J. Opt. Soc. Am. B 31, A20–A23 (2014).
    [Crossref]
  85. https://github.com/JPinnell/Modal-analysis-of-structured-light .
  86. G. Vallone, “Role of beam waist in Laguerre–Gauss expansion of vortex beams,” Opt. Lett. 42, 1097–1100 (2017).
    [Crossref]
  87. I. Nape, B. Sephton, Y.-W. Huang, A. Vallés, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “Enhancing the modal purity of orbital angular momentum photons,” APL Photon. 5, 070802 (2020).
    [Crossref]
  88. B. Sephton, A. Dudley, and A. Forbes, “Revealing the radial modes in vortex beams,” Appl. Opt. 55, 7830–7835 (2016).
    [Crossref]
  89. C. Schulze, S. Ngcobo, M. Duparré, and A. Forbes, “Modal decomposition without a priori scale information,” Opt. Express 20, 27866–27873 (2012).
    [Crossref]
  90. A. Jesacher, A. Schwaighofer, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express 15, 5801–5808 (2007).
    [Crossref]
  91. A. Forbes, Laser Beam Propagation: Generation and Propagation of Customized Light (CRC Press, 2014).

2020 (1)

I. Nape, B. Sephton, Y.-W. Huang, A. Vallés, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “Enhancing the modal purity of orbital angular momentum photons,” APL Photon. 5, 070802 (2020).
[Crossref]

2019 (9)

G. Lazarev, P.-J. Chen, J. Strauss, N. Fontaine, and A. Forbes, “Beyond the display: phase-only liquid crystal on silicon devices and their applications in photonics,” Opt. Express 27, 16206–16249 (2019).
[Crossref]

M. A. Cox, E. Toninelli, L. Cheng, M. J. Padgett, and A. Forbes, “A high-speed, wavelength invariant, single-pixel wavefront sensor with a digital micromirror device,” IEEE Access 7, 85860–85866 (2019).
[Crossref]

S. Scholes, R. Kara, J. Pinnell, V. Rodríguez-Fajardo, and A. Forbes, “Structured light with digital micromirror devices: a guide to best practice,” Opt. Eng. 59, 041202 (2019).
[Crossref]

A. Forbes and I. Nape, “Quantum mechanics with patterns of light: progress in high dimensional and multidimensional entanglement with structured light,” AVS Quantum Sci. 1, 011701 (2019).
[Crossref]

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10, 1865 (2019).
[Crossref]

G. Ruffato, M. Massari, M. Girardi, G. Parisi, M. Zontini, and F. Romanato, “Non-paraxial design and fabrication of a compact OAM sorter in the telecom infrared,” Opt. Express 27, 24123–24134 (2019).
[Crossref]

S. Pachava, R. Dharmavarapu, A. Vijayakumar, S. Jayakumar, A. Manthalkar, A. Dixit, N. K. Viswanathan, B. Srinivasan, and S. Bhattacharya, “Generation and decomposition of scalar and vector modes carrying orbital angular momentum: a review,” Opt. Eng. 59, 041205 (2019).
[Crossref]

A. Trichili, K.-H. Park, M. Zghal, B. S. Ooi, and M.-S. Alouini, “Communicating using spatial mode multiplexing: potentials, challenges and perspectives,” Commun. Surveys Tuts. 21, 3175–3203 (2019).
[Crossref]

K. Choi and C. Jun, “High-precision modal decomposition of laser beams based on globally optimized SPGD algorithm,” IEEE Photon. J. 11, 7103110 (2019).
[Crossref]

2018 (5)

2017 (7)

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

C. Rosales-Guzmán, N. Bhebhe, and A. Forbes, “Simultaneous generation of multiple vector beams on a single SLM,” Opt. Express 25, 25697–25706 (2017).
[Crossref]

B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. J. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep. 7, 13882 (2017).
[Crossref]

G. Kulkarni, R. Sahu, O. S. Magaña-Loaiza, R. W. Boyd, and A. K. Jha, “Single-shot measurement of the orbital-angular-momentum spectrum of light,” Nat. Commun. 8, 1054 (2017).
[Crossref]

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6, e16251 (2017).
[Crossref]

A. D’Errico, R. D’Amelio, B. Piccirillo, F. Cardano, and L. Marrucci, “Measuring the complex orbital angular momentum spectrum and spatial mode decomposition of structured light beams,” Optica 4, 1350–1357 (2017).
[Crossref]

G. Vallone, “Role of beam waist in Laguerre–Gauss expansion of vortex beams,” Opt. Lett. 42, 1097–1100 (2017).
[Crossref]

2016 (5)

2015 (2)

2014 (4)

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

A. Trichili, T. Mhlanga, Y. Ismail, F. S. Roux, M. McLaren, M. Zghal, and A. Forbes, “Detection of Bessel beams with digital axicons,” Opt. Express 22, 17553–17560 (2014).
[Crossref]

F. S. Roux and Y. Zhang, “Projective measurements in quantum and classical optical systems,” Phys. Rev. A 90, 033835 (2014).
[Crossref]

H. Qassim, F. M. Miatto, J. P. Torres, M. J. Padgett, E. Karimi, and R. W. Boyd, “Limitations to the determination of a Laguerre–Gauss spectrum via projective, phase-flattening measurement,” J. Opt. Soc. Am. B 31, A20–A23 (2014).
[Crossref]

2013 (6)

2012 (11)

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).
[Crossref]

C. Schulze, D. Flamm, M. Duparré, and A. Forbes, “Beam-quality measurements using a spatial light modulator,” Opt. Lett. 37, 4687–4689 (2012).
[Crossref]

C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20, 19714–19725 (2012).
[Crossref]

D. Flamm, C. Schulze, R. Brüning, O. A. Schmidt, T. Kaiser, S. Schröter, and M. Duparré, “Fast M2 measurement for fiber beams based on modal analysis,” Appl. Opt. 51, 987–993 (2012).
[Crossref]

A. Dudley, I. Litvin, and A. Forbes, “Quantitative measurement of the orbital angular momentum density of light,” Appl. Opt. 51, 823–833 (2012).
[Crossref]

G. Boudarham and M. Kociak, “Modal decompositions of the local electromagnetic density of states and spatially resolved electron energy loss probability in terms of geometric modes,” Phys. Rev. B 85, 245447 (2012).
[Crossref]

I. A. Litvin, A. Dudley, F. S. Roux, and A. Forbes, “Azimuthal decomposition with digital holograms,” Opt. Express 20, 10996–11004 (2012).
[Crossref]

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).
[Crossref]

D. Naidoo, K. At-Ameur, M. Brunel, and A. Forbes, “Intra-cavity generation of superpositions of Laguerre–Gaussian beams,” Appl. Phys. B 106, 683–690 (2012).
[Crossref]

M. Paurisse, L. Lévèque, M. Hanna, F. Druon, and P. Georges, “Complete measurement of fiber modal content by wavefront analysis,” Opt. Express 20, 4074–4084 (2012).
[Crossref]

C. Schulze, S. Ngcobo, M. Duparré, and A. Forbes, “Modal decomposition without a priori scale information,” Opt. Express 20, 27866–27873 (2012).
[Crossref]

2011 (5)

2010 (4)

J. Hickmann, E. Fonseca, W. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[Crossref]

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).
[Crossref]

Y. X. Ren, M. Li, K. Huang, J. G. Wu, H. F. Gao, Z. Q. Wang, and Y. M. Li, “Experimental generation of Laguerre-Gaussian beam using digital micromirror device,” Appl. Opt. 49, 1838–1844 (2010).
[Crossref]

D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010).
[Crossref]

2009 (5)

E. J. R. Vesseur, F. J. García de Abajo, and A. Polman, “Modal decomposition of surface–plasmon whispering gallery resonators,” Nano Lett. 9, 3147–3150 (2009).
[Crossref]

G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801 (2009).
[Crossref]

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).
[Crossref]

B. R. Boruah, “Dynamic manipulation of a laser beam using a liquid crystal spatial light modulator,” Am. J. Phys. 77, 331–336 (2009).
[Crossref]

M. Paurisse, M. Hanna, F. Druon, P. Georges, C. Bellanger, A. Brignon, and J. Huignard, “Phase and amplitude control of a multimode LMA fiber beam by use of digital holography,” Opt. Express 17, 13000–13008 (2009).
[Crossref]

2007 (2)

2006 (2)

2005 (2)

M. Duparré, B. Lüdge, and S. Schröter, “On-line characterization of Nd:YAG laser beams by means of modal decomposition using diffractive optical correlation filters,” Proc. SPIE 5962, 755–764 (2005).
[Crossref]

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94, 143902 (2005).
[Crossref]

2004 (2)

2003 (1)

M. R. Duparré, C. Rockstuhl, A. Letsch, S. Schroeter, and V. S. Pavelyev, “On-line control of laser beam quality by means of diffractive optical components,” Proc. SPIE 4932, 549–559 (2003).
[Crossref]

2002 (1)

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).
[Crossref]

2001 (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]

2000 (1)

M. R. Duparré, V. S. Pavelyev, V. A. Soifer, and B. Luedge, “Laser beam characterization by means of diffractive optical correlation filters,” Proc. SPIE 4095, 40–48 (2000).
[Crossref]

1998 (1)

M. R. Duparré, V. S. Pavelyev, B. Luedge, E.-B. Kley, V. A. Soifer, and R. M. Kowarschik, “Generation, superposition, and separation of Gauss-Hermite modes by means of does,” Proc. SPIE 3291, 104–114 (1998).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref]

1991 (2)

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[Crossref]

M. A. Golub, I. N. Sisakian, V. A. Soifer, and G. V. Uvarov, “New measurement techniques for modal power distribution in fibers,” Proc. SPIE 1366, 273–282 (1991).
[Crossref]

1987 (2)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

T. Yang, “A method of range and depth estimation by modal decomposition,” J. Acoust. Soc. Am. 82, 1736–1745 (1987).
[Crossref]

1976 (1)

R. Lindzen, “A modal decomposition of the semidiurnal tide in the lower thermosphere,” J. Geophys. Res. 81, 2923–2926 (1976).
[Crossref]

1973 (1)

1971 (1)

Abouraddy, A. F.

D. Mardani, A. F. Abouraddy, and G. K. Atia, “Interferometry-based modal analysis with finite aperture effects,” J. Opt. Soc. Am. A 35, 1880–1890 (2018).
[Crossref]

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94, 143902 (2005).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref]

Alouini, M.-S.

A. Trichili, K.-H. Park, M. Zghal, B. S. Ooi, and M.-S. Alouini, “Communicating using spatial mode multiplexing: potentials, challenges and perspectives,” Commun. Surveys Tuts. 21, 3175–3203 (2019).
[Crossref]

Alpmann, C.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Ambrosio, A.

I. Nape, B. Sephton, Y.-W. Huang, A. Vallés, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “Enhancing the modal purity of orbital angular momentum photons,” APL Photon. 5, 070802 (2020).
[Crossref]

Andersen, G. P.

G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801 (2009).
[Crossref]

Andrews, D. L.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Arnold, A. S.

Arrizón, V.

At-Ameur, K.

D. Naidoo, K. At-Ameur, M. Brunel, and A. Forbes, “Intra-cavity generation of superpositions of Laguerre–Gaussian beams,” Appl. Phys. B 106, 683–690 (2012).
[Crossref]

Atia, G. K.

Baker, M.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Bandres, M. A.

Banzer, P.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Barnett, S. M.

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).
[Crossref]

Bartelt, H.

Bauer, T.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Baumgartl, J.

Beijersbergen, M. W.

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).
[Crossref]

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref]

Bellanger, C.

Belmonte, A.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Bentley, J. B.

Berkhout, G. C. G.

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).
[Crossref]

Bernet, S.

Berry, M. V.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Bhattacharya, S.

S. Pachava, R. Dharmavarapu, A. Vijayakumar, S. Jayakumar, A. Manthalkar, A. Dixit, N. K. Viswanathan, B. Srinivasan, and S. Bhattacharya, “Generation and decomposition of scalar and vector modes carrying orbital angular momentum: a review,” Opt. Eng. 59, 041205 (2019).
[Crossref]

Bhebhe, N.

Bigelow, N. P.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Borchardt, J.

Boruah, B. R.

B. R. Boruah, “Dynamic manipulation of a laser beam using a liquid crystal spatial light modulator,” Am. J. Phys. 77, 331–336 (2009).
[Crossref]

B. R. Boruah and M. A. Neil, “Susceptibility to and correction of azimuthal aberrations in singular light beams,” Opt. Express 14, 10377–10385 (2006).
[Crossref]

Bouchard, F.

Boudarham, G.

G. Boudarham and M. Kociak, “Modal decompositions of the local electromagnetic density of states and spatially resolved electron energy loss probability in terms of geometric modes,” Phys. Rev. B 85, 245447 (2012).
[Crossref]

Boyd, R. W.

Brandt, F.

Brignon, A.

Brunel, M.

D. Naidoo, K. At-Ameur, M. Brunel, and A. Forbes, “Intra-cavity generation of superpositions of Laguerre–Gaussian beams,” Appl. Phys. B 106, 683–690 (2012).
[Crossref]

Brüning, R.

Cai, X.-L.

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6, e16251 (2017).
[Crossref]

Capaldo, P.

G. Ruffato, M. Girardi, M. Massari, E. Mafakheri, B. Sephton, P. Capaldo, A. Forbes, and F. Romanato, “A compact diffractive sorter for high-resolution demultiplexing of orbital angular momentum beams,” Sci. Rep. 8, 10248 (2018).
[Crossref]

Capasso, F.

I. Nape, B. Sephton, Y.-W. Huang, A. Vallés, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “Enhancing the modal purity of orbital angular momentum photons,” APL Photon. 5, 070802 (2020).
[Crossref]

Cardano, F.

Carpenter, J.

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10, 1865 (2019).
[Crossref]

Carrada, R.

Changhai, L.

Chávez-Cerda, S.

J. Hickmann, E. Fonseca, W. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[Crossref]

Chen, C.

Chen, D.-X.

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6, e16251 (2017).
[Crossref]

Chen, H.

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10, 1865 (2019).
[Crossref]

Chen, K.

G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801 (2009).
[Crossref]

Chen, P.-J.

Cheng, L.

M. A. Cox, E. Toninelli, L. Cheng, M. J. Padgett, and A. Forbes, “A high-speed, wavelength invariant, single-pixel wavefront sensor with a digital micromirror device,” IEEE Access 7, 85860–85866 (2019).
[Crossref]

Choi, K.

K. Choi and C. Jun, “High-precision modal decomposition of laser beams based on globally optimized SPGD algorithm,” IEEE Photon. J. 11, 7103110 (2019).
[Crossref]

Clark, T. W.

Courtial, J.

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

Cox, M. A.

M. A. Cox, E. Toninelli, L. Cheng, M. J. Padgett, and A. Forbes, “A high-speed, wavelength invariant, single-pixel wavefront sensor with a digital micromirror device,” IEEE Access 7, 85860–85866 (2019).
[Crossref]

B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Lightwave Technol. 36, 292–301 (2018).
[Crossref]

D’Amelio, R.

D’Errico, A.

Davis, J. A.

Dennis, M. R.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Denz, C.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Dharmavarapu, R.

S. Pachava, R. Dharmavarapu, A. Vijayakumar, S. Jayakumar, A. Manthalkar, A. Dixit, N. K. Viswanathan, B. Srinivasan, and S. Bhattacharya, “Generation and decomposition of scalar and vector modes carrying orbital angular momentum: a review,” Opt. Eng. 59, 041205 (2019).
[Crossref]

Dholakia, K.

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).
[Crossref]

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).
[Crossref]

Dixit, A.

S. Pachava, R. Dharmavarapu, A. Vijayakumar, S. Jayakumar, A. Manthalkar, A. Dixit, N. K. Viswanathan, B. Srinivasan, and S. Bhattacharya, “Generation and decomposition of scalar and vector modes carrying orbital angular momentum: a review,” Opt. Eng. 59, 041205 (2019).
[Crossref]

Dong, J.-J.

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6, e16251 (2017).
[Crossref]

Druon, F.

Dudley, A.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

B. Sephton, A. Dudley, and A. Forbes, “Revealing the radial modes in vortex beams,” Appl. Opt. 55, 7830–7835 (2016).
[Crossref]

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photon. 8, 200–227 (2016).
[Crossref]

A. Dudley, T. Mhlanga, M. Lavery, A. McDonald, F. S. Roux, M. Padgett, and A. Forbes, “Efficient sorting of Bessel beams,” Opt. Express 21, 165–171 (2013).
[Crossref]

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Reconstruction of laser beam wavefronts based on mode analysis,” Appl. Opt. 52, 5312–5317 (2013).
[Crossref]

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[Crossref]

A. Dudley, I. Litvin, and A. Forbes, “Quantitative measurement of the orbital angular momentum density of light,” Appl. Opt. 51, 823–833 (2012).
[Crossref]

I. A. Litvin, A. Dudley, F. S. Roux, and A. Forbes, “Azimuthal decomposition with digital holograms,” Opt. Express 20, 10996–11004 (2012).
[Crossref]

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).
[Crossref]

Duparré, M.

D. Flamm, C. Schulze, D. Naidoo, S. Schroter, 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]

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[Crossref]

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Reconstruction of laser beam wavefronts based on mode analysis,” Appl. Opt. 52, 5312–5317 (2013).
[Crossref]

C. Schulze, A. Lorenz, D. Flamm, A. Hartung, S. Schröter, H. Bartelt, and M. Duparré, “Mode resolved bend loss in few-mode optical fibers,” Opt. Express 21, 3170–3181 (2013).
[Crossref]

C. Schulze, S. Ngcobo, M. Duparré, and A. Forbes, “Modal decomposition without a priori scale information,” Opt. Express 20, 27866–27873 (2012).
[Crossref]

D. Flamm, C. Schulze, R. Brüning, O. A. Schmidt, T. Kaiser, S. Schröter, and M. Duparré, “Fast M2 measurement for fiber beams based on modal analysis,” Appl. Opt. 51, 987–993 (2012).
[Crossref]

C. Schulze, D. Flamm, M. Duparré, and A. Forbes, “Beam-quality measurements using a spatial light modulator,” Opt. Lett. 37, 4687–4689 (2012).
[Crossref]

C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20, 19714–19725 (2012).
[Crossref]

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).
[Crossref]

O. A. Schmidt, C. Schulze, D. Flamm, R. Brüning, T. Kaiser, S. Schröter, and M. Duparré, “Real-time determination of laser beam quality by modal decomposition,” Opt. Express 19, 6741–6748 (2011).
[Crossref]

D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010).
[Crossref]

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).
[Crossref]

M. Duparré, B. Lüdge, and S. Schröter, “On-line characterization of Nd:YAG laser beams by means of modal decomposition using diffractive optical correlation filters,” Proc. SPIE 5962, 755–764 (2005).
[Crossref]

Duparré, M. R.

M. R. Duparré, C. Rockstuhl, A. Letsch, S. Schroeter, and V. S. Pavelyev, “On-line control of laser beam quality by means of diffractive optical components,” Proc. SPIE 4932, 549–559 (2003).
[Crossref]

M. R. Duparré, V. S. Pavelyev, V. A. Soifer, and B. Luedge, “Laser beam characterization by means of diffractive optical correlation filters,” Proc. SPIE 4095, 40–48 (2000).
[Crossref]

M. R. Duparré, V. S. Pavelyev, B. Luedge, E.-B. Kley, V. A. Soifer, and R. M. Kowarschik, “Generation, superposition, and separation of Gauss-Hermite modes by means of does,” Proc. SPIE 3291, 104–114 (1998).
[Crossref]

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Dussan, L.

G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801 (2009).
[Crossref]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Fengjie, X.

Fickler, R.

F. Bouchard, N. H. Valencia, F. Brandt, R. Fickler, M. Huber, and M. Malik, “Measuring azimuthal and radial modes of photons,” Opt. Express 26, 31925–31941 (2018).
[Crossref]

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Fink, Y.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94, 143902 (2005).
[Crossref]

Flamm, D.

D. Flamm, C. Schulze, D. Naidoo, S. Schroter, 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]

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[Crossref]

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Reconstruction of laser beam wavefronts based on mode analysis,” Appl. Opt. 52, 5312–5317 (2013).
[Crossref]

C. Schulze, A. Lorenz, D. Flamm, A. Hartung, S. Schröter, H. Bartelt, and M. Duparré, “Mode resolved bend loss in few-mode optical fibers,” Opt. Express 21, 3170–3181 (2013).
[Crossref]

D. Flamm, C. Schulze, R. Brüning, O. A. Schmidt, T. Kaiser, S. Schröter, and M. Duparré, “Fast M2 measurement for fiber beams based on modal analysis,” Appl. Opt. 51, 987–993 (2012).
[Crossref]

C. Schulze, D. Flamm, M. Duparré, and A. Forbes, “Beam-quality measurements using a spatial light modulator,” Opt. Lett. 37, 4687–4689 (2012).
[Crossref]

C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20, 19714–19725 (2012).
[Crossref]

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).
[Crossref]

O. A. Schmidt, C. Schulze, D. Flamm, R. Brüning, T. Kaiser, S. Schröter, and M. Duparré, “Real-time determination of laser beam quality by modal decomposition,” Opt. Express 19, 6741–6748 (2011).
[Crossref]

D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010).
[Crossref]

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).
[Crossref]

Fonseca, E.

J. Hickmann, E. Fonseca, W. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[Crossref]

Fontaine, N.

Fontaine, N. K.

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10, 1865 (2019).
[Crossref]

Forbes, A.

I. Nape, B. Sephton, Y.-W. Huang, A. Vallés, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “Enhancing the modal purity of orbital angular momentum photons,” APL Photon. 5, 070802 (2020).
[Crossref]

G. Lazarev, P.-J. Chen, J. Strauss, N. Fontaine, and A. Forbes, “Beyond the display: phase-only liquid crystal on silicon devices and their applications in photonics,” Opt. Express 27, 16206–16249 (2019).
[Crossref]

S. Scholes, R. Kara, J. Pinnell, V. Rodríguez-Fajardo, and A. Forbes, “Structured light with digital micromirror devices: a guide to best practice,” Opt. Eng. 59, 041202 (2019).
[Crossref]

M. A. Cox, E. Toninelli, L. Cheng, M. J. Padgett, and A. Forbes, “A high-speed, wavelength invariant, single-pixel wavefront sensor with a digital micromirror device,” IEEE Access 7, 85860–85866 (2019).
[Crossref]

A. Forbes and I. Nape, “Quantum mechanics with patterns of light: progress in high dimensional and multidimensional entanglement with structured light,” AVS Quantum Sci. 1, 011701 (2019).
[Crossref]

B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Lightwave Technol. 36, 292–301 (2018).
[Crossref]

G. Ruffato, M. Girardi, M. Massari, E. Mafakheri, B. Sephton, P. Capaldo, A. Forbes, and F. Romanato, “A compact diffractive sorter for high-resolution demultiplexing of orbital angular momentum beams,” Sci. Rep. 8, 10248 (2018).
[Crossref]

B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. J. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep. 7, 13882 (2017).
[Crossref]

C. Rosales-Guzmán, N. Bhebhe, and A. Forbes, “Simultaneous generation of multiple vector beams on a single SLM,” Opt. Express 25, 25697–25706 (2017).
[Crossref]

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

A. Forbes, A. Dudley, and M. McLaren, “Creation and detection of optical modes with spatial light modulators,” Adv. Opt. Photon. 8, 200–227 (2016).
[Crossref]

B. Ndagano, H. Sroor, M. McLaren, C. Rosales-Guzmán, and A. Forbes, “Beam quality measure for vector beams,” Opt. Lett. 41, 3407–3410 (2016).
[Crossref]

B. Sephton, A. Dudley, and A. Forbes, “Revealing the radial modes in vortex beams,” Appl. Opt. 55, 7830–7835 (2016).
[Crossref]

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
[Crossref]

A. Trichili, T. Mhlanga, Y. Ismail, F. S. Roux, M. McLaren, M. Zghal, and A. Forbes, “Detection of Bessel beams with digital axicons,” Opt. Express 22, 17553–17560 (2014).
[Crossref]

D. Flamm, C. Schulze, D. Naidoo, S. Schroter, 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]

A. Dudley, T. Mhlanga, M. Lavery, A. McDonald, F. S. Roux, M. Padgett, and A. Forbes, “Efficient sorting of Bessel beams,” Opt. Express 21, 165–171 (2013).
[Crossref]

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Reconstruction of laser beam wavefronts based on mode analysis,” Appl. Opt. 52, 5312–5317 (2013).
[Crossref]

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[Crossref]

C. Schulze, D. Flamm, M. Duparré, and A. Forbes, “Beam-quality measurements using a spatial light modulator,” Opt. Lett. 37, 4687–4689 (2012).
[Crossref]

A. Dudley, I. Litvin, and A. Forbes, “Quantitative measurement of the orbital angular momentum density of light,” Appl. Opt. 51, 823–833 (2012).
[Crossref]

C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20, 19714–19725 (2012).
[Crossref]

D. Naidoo, K. At-Ameur, M. Brunel, and A. Forbes, “Intra-cavity generation of superpositions of Laguerre–Gaussian beams,” Appl. Phys. B 106, 683–690 (2012).
[Crossref]

C. Schulze, S. Ngcobo, M. Duparré, and A. Forbes, “Modal decomposition without a priori scale information,” Opt. Express 20, 27866–27873 (2012).
[Crossref]

I. A. Litvin, A. Dudley, F. S. Roux, and A. Forbes, “Azimuthal decomposition with digital holograms,” Opt. Express 20, 10996–11004 (2012).
[Crossref]

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).
[Crossref]

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).
[Crossref]

C. Rosales-Guzmán and A. Forbes, How to Shape Light with Spatial Light Modulators (SPIE, 2017).

A. Forbes, Laser Beam Propagation: Generation and Propagation of Customized Light (CRC Press, 2014).

Franke-Arnold, S.

T. W. Clark, R. F. Offer, S. Franke-Arnold, A. S. Arnold, and N. Radwell, “Comparison of beam generation techniques using a phase only spatial light modulator,” Opt. Express 24, 6249–6264 (2016).
[Crossref]

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).
[Crossref]

Fu, D.-Z.

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6, e16251 (2017).
[Crossref]

Fürhapter, S.

Gaida, C.

Gao, H. F.

García de Abajo, F. J.

E. J. R. Vesseur, F. J. García de Abajo, and A. Polman, “Modal decomposition of surface–plasmon whispering gallery resonators,” Nano Lett. 9, 3147–3150 (2009).
[Crossref]

Georges, P.

Ghebremichael, F.

G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801 (2009).
[Crossref]

Girardi, M.

G. Ruffato, M. Massari, M. Girardi, G. Parisi, M. Zontini, and F. Romanato, “Non-paraxial design and fabrication of a compact OAM sorter in the telecom infrared,” Opt. Express 27, 24123–24134 (2019).
[Crossref]

G. Ruffato, M. Girardi, M. Massari, E. Mafakheri, B. Sephton, P. Capaldo, A. Forbes, and F. Romanato, “A compact diffractive sorter for high-resolution demultiplexing of orbital angular momentum beams,” Sci. Rep. 8, 10248 (2018).
[Crossref]

Golub, M. A.

M. A. Golub, I. N. Sisakian, V. A. Soifer, and G. V. Uvarov, “New measurement techniques for modal power distribution in fibers,” Proc. SPIE 1366, 273–282 (1991).
[Crossref]

V. A. Soifer and M. A. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).

González, L. A.

Gordon, R.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Gutiérrez-Vega, J. C.

Hanna, M.

Hartung, A.

Hernandez-Aranda, R. I.

B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. J. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep. 7, 13882 (2017).
[Crossref]

Hickmann, J.

J. Hickmann, E. Fonseca, W. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[Crossref]

Huang, K.

Huang, Y.-W.

I. Nape, B. Sephton, Y.-W. Huang, A. Vallés, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “Enhancing the modal purity of orbital angular momentum photons,” APL Photon. 5, 070802 (2020).
[Crossref]

Huber, M.

Huignard, J.

Ismail, Y.

Jansen, F.

Jauregui, C.

Jayakumar, S.

S. Pachava, R. Dharmavarapu, A. Vijayakumar, S. Jayakumar, A. Manthalkar, A. Dixit, N. K. Viswanathan, B. Srinivasan, and S. Bhattacharya, “Generation and decomposition of scalar and vector modes carrying orbital angular momentum: a review,” Opt. Eng. 59, 041205 (2019).
[Crossref]

Jesacher, A.

Jha, A. K.

G. Kulkarni, R. Sahu, O. S. Magaña-Loaiza, R. W. Boyd, and A. K. Jha, “Single-shot measurement of the orbital-angular-momentum spectrum of light,” Nat. Commun. 8, 1054 (2017).
[Crossref]

Joannopoulos, J. D.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94, 143902 (2005).
[Crossref]

Jones, A. L.

Jun, C.

K. Choi and C. Jun, “High-precision modal decomposition of laser beams based on globally optimized SPGD algorithm,” IEEE Photon. J. 11, 7103110 (2019).
[Crossref]

Kaiser, T.

Kara, R.

S. Scholes, R. Kara, J. Pinnell, V. Rodríguez-Fajardo, and A. Forbes, “Structured light with digital micromirror devices: a guide to best practice,” Opt. Eng. 59, 041202 (2019).
[Crossref]

Karimi, E.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

H. Qassim, F. M. Miatto, J. P. Torres, M. J. Padgett, E. Karimi, and R. W. Boyd, “Limitations to the determination of a Laguerre–Gauss spectrum via projective, phase-flattening measurement,” J. Opt. Soc. Am. B 31, A20–A23 (2014).
[Crossref]

Kim, K.

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10, 1865 (2019).
[Crossref]

Kirk, J. P.

Kley, E.-B.

M. R. Duparré, V. S. Pavelyev, B. Luedge, E.-B. Kley, V. A. Soifer, and R. M. Kowarschik, “Generation, superposition, and separation of Gauss-Hermite modes by means of does,” Proc. SPIE 3291, 104–114 (1998).
[Crossref]

Kociak, M.

G. Boudarham and M. Kociak, “Modal decompositions of the local electromagnetic density of states and spatially resolved electron energy loss probability in terms of geometric modes,” Phys. Rev. B 85, 245447 (2012).
[Crossref]

Konrad, T.

B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. J. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep. 7, 13882 (2017).
[Crossref]

M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
[Crossref]

Kowarschik, R. M.

M. R. Duparré, V. S. Pavelyev, B. Luedge, E.-B. Kley, V. A. Soifer, and R. M. Kowarschik, “Generation, superposition, and separation of Gauss-Hermite modes by means of does,” Proc. SPIE 3291, 104–114 (1998).
[Crossref]

Kulkarni, G.

G. Kulkarni, R. Sahu, O. S. Magaña-Loaiza, R. W. Boyd, and A. K. Jha, “Single-shot measurement of the orbital-angular-momentum spectrum of light,” Nat. Commun. 8, 1054 (2017).
[Crossref]

Lavery, M.

Lavery, M. P.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

Lavery, M. P. J.

B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. J. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep. 7, 13882 (2017).
[Crossref]

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).
[Crossref]

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).
[Crossref]

Lazarev, G.

Leach, J.

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

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).
[Crossref]

Letsch, A.

M. R. Duparré, C. Rockstuhl, A. Letsch, S. Schroeter, and V. S. Pavelyev, “On-line control of laser beam quality by means of diffractive optical components,” Proc. SPIE 4932, 549–559 (2003).
[Crossref]

Lévèque, L.

Li, F.-L.

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6, e16251 (2017).
[Crossref]

Li, M.

Li, Y. M.

Limpert, J.

Lindzen, R.

R. Lindzen, “A modal decomposition of the semidiurnal tide in the lower thermosphere,” J. Geophys. Res. 81, 2923–2926 (1976).
[Crossref]

Litchinitser, N. M.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Litvin, I.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

A. Dudley, I. Litvin, and A. Forbes, “Quantitative measurement of the orbital angular momentum density of light,” Appl. Opt. 51, 823–833 (2012).
[Crossref]

Litvin, I. A.

Lorenz, A.

Lüdge, B.

M. Duparré, B. Lüdge, and S. Schröter, “On-line characterization of Nd:YAG laser beams by means of modal decomposition using diffractive optical correlation filters,” Proc. SPIE 5962, 755–764 (2005).
[Crossref]

Luedge, B.

M. R. Duparré, V. S. Pavelyev, V. A. Soifer, and B. Luedge, “Laser beam characterization by means of diffractive optical correlation filters,” Proc. SPIE 4095, 40–48 (2000).
[Crossref]

M. R. Duparré, V. S. Pavelyev, B. Luedge, E.-B. Kley, V. A. Soifer, and R. M. Kowarschik, “Generation, superposition, and separation of Gauss-Hermite modes by means of does,” Proc. SPIE 3291, 104–114 (1998).
[Crossref]

Mafakheri, E.

G. Ruffato, M. Girardi, M. Massari, E. Mafakheri, B. Sephton, P. Capaldo, A. Forbes, and F. Romanato, “A compact diffractive sorter for high-resolution demultiplexing of orbital angular momentum beams,” Sci. Rep. 8, 10248 (2018).
[Crossref]

Magaña-Loaiza, O. S.

G. Kulkarni, R. Sahu, O. S. Magaña-Loaiza, R. W. Boyd, and A. K. Jha, “Single-shot measurement of the orbital-angular-momentum spectrum of light,” Nat. Commun. 8, 1054 (2017).
[Crossref]

M. Mirhosseini, O. S. Magaña-Loaiza, C. Chen, B. Rodenburg, M. Malik, and R. W. Boyd, “Rapid generation of light beams carrying orbital angular momentum,” Opt. Express 21, 30196–30203 (2013).
[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.

Mansuripur, M.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Manthalkar, A.

S. Pachava, R. Dharmavarapu, A. Vijayakumar, S. Jayakumar, A. Manthalkar, A. Dixit, N. K. Viswanathan, B. Srinivasan, and S. Bhattacharya, “Generation and decomposition of scalar and vector modes carrying orbital angular momentum: a review,” Opt. Eng. 59, 041205 (2019).
[Crossref]

Mardani, D.

Margiewicz, J.

Marrucci, L.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

A. D’Errico, R. D’Amelio, B. Piccirillo, F. Cardano, and L. Marrucci, “Measuring the complex orbital angular momentum spectrum and spatial mode decomposition of structured light beams,” Optica 4, 1350–1357 (2017).
[Crossref]

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

Massari, M.

G. Ruffato, M. Massari, M. Girardi, G. Parisi, M. Zontini, and F. Romanato, “Non-paraxial design and fabrication of a compact OAM sorter in the telecom infrared,” Opt. Express 27, 24123–24134 (2019).
[Crossref]

G. Ruffato, M. Girardi, M. Massari, E. Mafakheri, B. Sephton, P. Capaldo, A. Forbes, and F. Romanato, “A compact diffractive sorter for high-resolution demultiplexing of orbital angular momentum beams,” Sci. Rep. 8, 10248 (2018).
[Crossref]

Maurer, C.

Mazilu, M.

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).
[Crossref]

McDonald, A.

McLaren, M.

McMorran, B.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Mhlanga, T.

Miatto, F. M.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Mirhosseini, M.

Mourka, A.

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).
[Crossref]

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).
[Crossref]

Naidoo, D.

Nape, I.

I. Nape, B. Sephton, Y.-W. Huang, A. Vallés, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “Enhancing the modal purity of orbital angular momentum photons,” APL Photon. 5, 070802 (2020).
[Crossref]

A. Forbes and I. Nape, “Quantum mechanics with patterns of light: progress in high dimensional and multidimensional entanglement with structured light,” AVS Quantum Sci. 1, 011701 (2019).
[Crossref]

B. Ndagano, I. Nape, M. A. Cox, C. Rosales-Guzman, and A. Forbes, “Creation and detection of vector vortex modes for classical and quantum communication,” J. Lightwave Technol. 36, 292–301 (2018).
[Crossref]

B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. J. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep. 7, 13882 (2017).
[Crossref]

Ndagano, B.

Neely, T. W.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Neil, M. A.

Neilson, D. T.

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10, 1865 (2019).
[Crossref]

Ngcobo, S.

Offer, R. F.

Ooi, B. S.

A. Trichili, K.-H. Park, M. Zghal, B. S. Ooi, and M.-S. Alouini, “Communicating using spatial mode multiplexing: potentials, challenges and perspectives,” Commun. Surveys Tuts. 21, 3175–3203 (2019).
[Crossref]

Otto, H.

Pachava, S.

S. Pachava, R. Dharmavarapu, A. Vijayakumar, S. Jayakumar, A. Manthalkar, A. Dixit, N. K. Viswanathan, B. Srinivasan, and S. Bhattacharya, “Generation and decomposition of scalar and vector modes carrying orbital angular momentum: a review,” Opt. Eng. 59, 041205 (2019).
[Crossref]

Padgett, M.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

A. Dudley, T. Mhlanga, M. Lavery, A. McDonald, F. S. Roux, M. Padgett, and A. Forbes, “Efficient sorting of Bessel beams,” Opt. Express 21, 165–171 (2013).
[Crossref]

Padgett, M. J.

M. A. Cox, E. Toninelli, L. Cheng, M. J. Padgett, and A. Forbes, “A high-speed, wavelength invariant, single-pixel wavefront sensor with a digital micromirror device,” IEEE Access 7, 85860–85866 (2019).
[Crossref]

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

H. Qassim, F. M. Miatto, J. P. Torres, M. J. Padgett, E. Karimi, and R. W. Boyd, “Limitations to the determination of a Laguerre–Gauss spectrum via projective, phase-flattening measurement,” J. Opt. Soc. Am. B 31, A20–A23 (2014).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

Parisi, G.

Park, K.-H.

A. Trichili, K.-H. Park, M. Zghal, B. S. Ooi, and M.-S. Alouini, “Communicating using spatial mode multiplexing: potentials, challenges and perspectives,” Commun. Surveys Tuts. 21, 3175–3203 (2019).
[Crossref]

Paurisse, M.

Pavelyev, V. S.

M. R. Duparré, C. Rockstuhl, A. Letsch, S. Schroeter, and V. S. Pavelyev, “On-line control of laser beam quality by means of diffractive optical components,” Proc. SPIE 4932, 549–559 (2003).
[Crossref]

M. R. Duparré, V. S. Pavelyev, V. A. Soifer, and B. Luedge, “Laser beam characterization by means of diffractive optical correlation filters,” Proc. SPIE 4095, 40–48 (2000).
[Crossref]

M. R. Duparré, V. S. Pavelyev, B. Luedge, E.-B. Kley, V. A. Soifer, and R. M. Kowarschik, “Generation, superposition, and separation of Gauss-Hermite modes by means of does,” Proc. SPIE 3291, 104–114 (1998).
[Crossref]

Perez-Garcia, B.

B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. J. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep. 7, 13882 (2017).
[Crossref]

Piccirillo, B.

A. D’Errico, R. D’Amelio, B. Piccirillo, F. Cardano, and L. Marrucci, “Measuring the complex orbital angular momentum spectrum and spatial mode decomposition of structured light beams,” Optica 4, 1350–1357 (2017).
[Crossref]

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[Crossref]

Pinnell, J.

S. Scholes, R. Kara, J. Pinnell, V. Rodríguez-Fajardo, and A. Forbes, “Structured light with digital micromirror devices: a guide to best practice,” Opt. Eng. 59, 041202 (2019).
[Crossref]

Polman, A.

E. J. R. Vesseur, F. J. García de Abajo, and A. Polman, “Modal decomposition of surface–plasmon whispering gallery resonators,” Nano Lett. 9, 3147–3150 (2009).
[Crossref]

Qassim, H.

Qiu, C.-W.

I. Nape, B. Sephton, Y.-W. Huang, A. Vallés, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “Enhancing the modal purity of orbital angular momentum photons,” APL Photon. 5, 070802 (2020).
[Crossref]

Radwell, N.

Rafsanjani, S. M. H.

Ren, Y. X.

Ritsch-Marte, M.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

A. Jesacher, A. Schwaighofer, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express 15, 5801–5808 (2007).
[Crossref]

Rivera, F.

Rockstuhl, C.

M. R. Duparré, C. Rockstuhl, A. Letsch, S. Schroeter, and V. S. Pavelyev, “On-line control of laser beam quality by means of diffractive optical components,” Proc. SPIE 4932, 549–559 (2003).
[Crossref]

Rodenburg, B.

Rodríguez-Fajardo, V.

S. Scholes, R. Kara, J. Pinnell, V. Rodríguez-Fajardo, and A. Forbes, “Structured light with digital micromirror devices: a guide to best practice,” Opt. Eng. 59, 041202 (2019).
[Crossref]

Romanato, F.

G. Ruffato, M. Massari, M. Girardi, G. Parisi, M. Zontini, and F. Romanato, “Non-paraxial design and fabrication of a compact OAM sorter in the telecom infrared,” Opt. Express 27, 24123–24134 (2019).
[Crossref]

G. Ruffato, M. Girardi, M. Massari, E. Mafakheri, B. Sephton, P. Capaldo, A. Forbes, and F. Romanato, “A compact diffractive sorter for high-resolution demultiplexing of orbital angular momentum beams,” Sci. Rep. 8, 10248 (2018).
[Crossref]

Romero, J.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Rosales-Guzman, C.

Rosales-Guzmán, C.

C. Rosales-Guzmán, N. Bhebhe, and A. Forbes, “Simultaneous generation of multiple vector beams on a single SLM,” Opt. Express 25, 25697–25706 (2017).
[Crossref]

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

B. Ndagano, H. Sroor, M. McLaren, C. Rosales-Guzmán, and A. Forbes, “Beam quality measure for vector beams,” Opt. Lett. 41, 3407–3410 (2016).
[Crossref]

C. Rosales-Guzmán and A. Forbes, How to Shape Light with Spatial Light Modulators (SPIE, 2017).

Roux, F. S.

Rubinsztein-Dunlop, H.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Ruffato, G.

G. Ruffato, M. Massari, M. Girardi, G. Parisi, M. Zontini, and F. Romanato, “Non-paraxial design and fabrication of a compact OAM sorter in the telecom infrared,” Opt. Express 27, 24123–24134 (2019).
[Crossref]

G. Ruffato, M. Girardi, M. Massari, E. Mafakheri, B. Sephton, P. Capaldo, A. Forbes, and F. Romanato, “A compact diffractive sorter for high-resolution demultiplexing of orbital angular momentum beams,” Sci. Rep. 8, 10248 (2018).
[Crossref]

Ruiz, U.

Ryf, R.

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10, 1865 (2019).
[Crossref]

Sahu, R.

G. Kulkarni, R. Sahu, O. S. Magaña-Loaiza, R. W. Boyd, and A. K. Jha, “Single-shot measurement of the orbital-angular-momentum spectrum of light,” Nat. Commun. 8, 1054 (2017).
[Crossref]

Schmidt, O. A.

Scholes, S.

S. Scholes, R. Kara, J. Pinnell, V. Rodríguez-Fajardo, and A. Forbes, “Structured light with digital micromirror devices: a guide to best practice,” Opt. Eng. 59, 041202 (2019).
[Crossref]

B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. J. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep. 7, 13882 (2017).
[Crossref]

Schroeter, S.

M. R. Duparré, C. Rockstuhl, A. Letsch, S. Schroeter, and V. S. Pavelyev, “On-line control of laser beam quality by means of diffractive optical components,” Proc. SPIE 4932, 549–559 (2003).
[Crossref]

Schroter, S.

Schröter, S.

Schulze, C.

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[Crossref]

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Reconstruction of laser beam wavefronts based on mode analysis,” Appl. Opt. 52, 5312–5317 (2013).
[Crossref]

C. Schulze, A. Lorenz, D. Flamm, A. Hartung, S. Schröter, H. Bartelt, and M. Duparré, “Mode resolved bend loss in few-mode optical fibers,” Opt. Express 21, 3170–3181 (2013).
[Crossref]

D. Flamm, C. Schulze, D. Naidoo, S. Schroter, 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]

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).
[Crossref]

C. Schulze, S. Ngcobo, M. Duparré, and A. Forbes, “Modal decomposition without a priori scale information,” Opt. Express 20, 27866–27873 (2012).
[Crossref]

D. Flamm, C. Schulze, R. Brüning, O. A. Schmidt, T. Kaiser, S. Schröter, and M. Duparré, “Fast M2 measurement for fiber beams based on modal analysis,” Appl. Opt. 51, 987–993 (2012).
[Crossref]

C. Schulze, D. Flamm, M. Duparré, and A. Forbes, “Beam-quality measurements using a spatial light modulator,” Opt. Lett. 37, 4687–4689 (2012).
[Crossref]

C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20, 19714–19725 (2012).
[Crossref]

O. A. Schmidt, C. Schulze, D. Flamm, R. Brüning, T. Kaiser, S. Schröter, and M. Duparré, “Real-time determination of laser beam quality by modal decomposition,” Opt. Express 19, 6741–6748 (2011).
[Crossref]

D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010).
[Crossref]

Schwaighofer, A.

Sephton, B.

I. Nape, B. Sephton, Y.-W. Huang, A. Vallés, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “Enhancing the modal purity of orbital angular momentum photons,” APL Photon. 5, 070802 (2020).
[Crossref]

G. Ruffato, M. Girardi, M. Massari, E. Mafakheri, B. Sephton, P. Capaldo, A. Forbes, and F. Romanato, “A compact diffractive sorter for high-resolution demultiplexing of orbital angular momentum beams,” Sci. Rep. 8, 10248 (2018).
[Crossref]

B. Sephton, A. Dudley, and A. Forbes, “Revealing the radial modes in vortex beams,” Appl. Opt. 55, 7830–7835 (2016).
[Crossref]

Shanor, C.

Shapira, O.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94, 143902 (2005).
[Crossref]

Shengyang, H.

Shi, Z.

Siegman, A. E.

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[Crossref]

A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
[Crossref]

A. E. Siegman, “Basic concepts,” in Lasers (University Science Books, 1986), Chap. 2, pp. 7–15.

A. E. Siegman, Defining and Measuring Laser Beam Quality (Springer, 1993), pp. 13–28.

Sisakian, I. N.

M. A. Golub, I. N. Sisakian, V. A. Soifer, and G. V. Uvarov, “New measurement techniques for modal power distribution in fibers,” Proc. SPIE 1366, 273–282 (1991).
[Crossref]

Soares, W.

J. Hickmann, E. Fonseca, W. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[Crossref]

Soifer, V. A.

M. R. Duparré, V. S. Pavelyev, V. A. Soifer, and B. Luedge, “Laser beam characterization by means of diffractive optical correlation filters,” Proc. SPIE 4095, 40–48 (2000).
[Crossref]

M. R. Duparré, V. S. Pavelyev, B. Luedge, E.-B. Kley, V. A. Soifer, and R. M. Kowarschik, “Generation, superposition, and separation of Gauss-Hermite modes by means of does,” Proc. SPIE 3291, 104–114 (1998).
[Crossref]

M. A. Golub, I. N. Sisakian, V. A. Soifer, and G. V. Uvarov, “New measurement techniques for modal power distribution in fibers,” Proc. SPIE 1366, 273–282 (1991).
[Crossref]

V. A. Soifer and M. A. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).

Spreeuw, R.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref]

Srinivasan, B.

S. Pachava, R. Dharmavarapu, A. Vijayakumar, S. Jayakumar, A. Manthalkar, A. Dixit, N. K. Viswanathan, B. Srinivasan, and S. Bhattacharya, “Generation and decomposition of scalar and vector modes carrying orbital angular momentum: a review,” Opt. Eng. 59, 041205 (2019).
[Crossref]

Sroor, H.

Stilgoe, A. B.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Strauss, J.

Stutzki, F.

Toninelli, E.

M. A. Cox, E. Toninelli, L. Cheng, M. J. Padgett, and A. Forbes, “A high-speed, wavelength invariant, single-pixel wavefront sensor with a digital micromirror device,” IEEE Access 7, 85860–85866 (2019).
[Crossref]

Torres, J. P.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

H. Qassim, F. M. Miatto, J. P. Torres, M. J. Padgett, E. Karimi, and R. W. Boyd, “Limitations to the determination of a Laguerre–Gauss spectrum via projective, phase-flattening measurement,” J. Opt. Soc. Am. B 31, A20–A23 (2014).
[Crossref]

Trichili, A.

A. Trichili, K.-H. Park, M. Zghal, B. S. Ooi, and M.-S. Alouini, “Communicating using spatial mode multiplexing: potentials, challenges and perspectives,” Commun. Surveys Tuts. 21, 3175–3203 (2019).
[Crossref]

A. Trichili, T. Mhlanga, Y. Ismail, F. S. Roux, M. McLaren, M. Zghal, and A. Forbes, “Detection of Bessel beams with digital axicons,” Opt. Express 22, 17553–17560 (2014).
[Crossref]

Tünnermann, A.

Uvarov, G. V.

M. A. Golub, I. N. Sisakian, V. A. Soifer, and G. V. Uvarov, “New measurement techniques for modal power distribution in fibers,” Proc. SPIE 1366, 273–282 (1991).
[Crossref]

Valencia, N. H.

Vallés, A.

I. Nape, B. Sephton, Y.-W. Huang, A. Vallés, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “Enhancing the modal purity of orbital angular momentum photons,” APL Photon. 5, 070802 (2020).
[Crossref]

Vallone, G.

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]

Vesseur, E. J. R.

E. J. R. Vesseur, F. J. García de Abajo, and A. Polman, “Modal decomposition of surface–plasmon whispering gallery resonators,” Nano Lett. 9, 3147–3150 (2009).
[Crossref]

Vettenburg, T.

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).
[Crossref]

Vijayakumar, A.

S. Pachava, R. Dharmavarapu, A. Vijayakumar, S. Jayakumar, A. Manthalkar, A. Dixit, N. K. Viswanathan, B. Srinivasan, and S. Bhattacharya, “Generation and decomposition of scalar and vector modes carrying orbital angular momentum: a review,” Opt. Eng. 59, 041205 (2019).
[Crossref]

Viswanathan, N. K.

S. Pachava, R. Dharmavarapu, A. Vijayakumar, S. Jayakumar, A. Manthalkar, A. Dixit, N. K. Viswanathan, B. Srinivasan, and S. Bhattacharya, “Generation and decomposition of scalar and vector modes carrying orbital angular momentum: a review,” Opt. Eng. 59, 041205 (2019).
[Crossref]

Wang, Z. Q.

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]

Weiner, A. M.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

White, A. G.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Willner, A. E.

Y. Zhou, J. Zhao, Z. Shi, S. M. H. Rafsanjani, M. Mirhosseini, Z. Zhu, A. E. Willner, and R. W. Boyd, “Hermite–Gaussian mode sorter,” Opt. Lett. 43, 5263–5266 (2018).
[Crossref]

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Woerdman, J.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref]

Wright, E. M.

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).
[Crossref]

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).
[Crossref]

Wu, J. G.

Xie, G.

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

Yang, T.

T. Yang, “A method of range and depth estimation by modal decomposition,” J. Acoust. Soc. Am. 82, 1736–1745 (1987).
[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]

Zghal, M.

A. Trichili, K.-H. Park, M. Zghal, B. S. Ooi, and M.-S. Alouini, “Communicating using spatial mode multiplexing: potentials, challenges and perspectives,” Commun. Surveys Tuts. 21, 3175–3203 (2019).
[Crossref]

A. Trichili, T. Mhlanga, Y. Ismail, F. S. Roux, M. McLaren, M. Zghal, and A. Forbes, “Detection of Bessel beams with digital axicons,” Opt. Express 22, 17553–17560 (2014).
[Crossref]

Zhang, P.

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6, e16251 (2017).
[Crossref]

Zhang, X.-L.

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6, e16251 (2017).
[Crossref]

Zhang, Y.

F. S. Roux and Y. Zhang, “Projective measurements in quantum and classical optical systems,” Phys. Rev. A 90, 033835 (2014).
[Crossref]

Zhao, J.

Zhou, H.-L.

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6, e16251 (2017).
[Crossref]

Zhou, Y.

Zhu, Z.

Zongfu, J.

Zontini, M.

Adv. Opt. Photon. (1)

Am. J. Phys. (1)

B. R. Boruah, “Dynamic manipulation of a laser beam using a liquid crystal spatial light modulator,” Am. J. Phys. 77, 331–336 (2009).
[Crossref]

APL Photon. (1)

I. Nape, B. Sephton, Y.-W. Huang, A. Vallés, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “Enhancing the modal purity of orbital angular momentum photons,” APL Photon. 5, 070802 (2020).
[Crossref]

Appl. Opt. (6)

Appl. Phys. B (1)

D. Naidoo, K. At-Ameur, M. Brunel, and A. Forbes, “Intra-cavity generation of superpositions of Laguerre–Gaussian beams,” Appl. Phys. B 106, 683–690 (2012).
[Crossref]

Appl. Phys. Lett. (1)

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).
[Crossref]

AVS Quantum Sci. (1)

A. Forbes and I. Nape, “Quantum mechanics with patterns of light: progress in high dimensional and multidimensional entanglement with structured light,” AVS Quantum Sci. 1, 011701 (2019).
[Crossref]

Commun. Surveys Tuts. (1)

A. Trichili, K.-H. Park, M. Zghal, B. S. Ooi, and M.-S. Alouini, “Communicating using spatial mode multiplexing: potentials, challenges and perspectives,” Commun. Surveys Tuts. 21, 3175–3203 (2019).
[Crossref]

IEEE Access (1)

M. A. Cox, E. Toninelli, L. Cheng, M. J. Padgett, and A. Forbes, “A high-speed, wavelength invariant, single-pixel wavefront sensor with a digital micromirror device,” IEEE Access 7, 85860–85866 (2019).
[Crossref]

IEEE J. Quantum Electron. (1)

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[Crossref]

IEEE Photon. J. (1)

K. Choi and C. Jun, “High-precision modal decomposition of laser beams based on globally optimized SPGD algorithm,” IEEE Photon. J. 11, 7103110 (2019).
[Crossref]

J. Acoust. Soc. Am. (1)

T. Yang, “A method of range and depth estimation by modal decomposition,” J. Acoust. Soc. Am. 82, 1736–1745 (1987).
[Crossref]

J. Geophys. Res. (1)

R. Lindzen, “A modal decomposition of the semidiurnal tide in the lower thermosphere,” J. Geophys. Res. 81, 2923–2926 (1976).
[Crossref]

J. Lightwave Technol. (2)

J. Opt. (1)

H. Rubinsztein-Dunlop, A. Forbes, M. V. Berry, M. R. Dennis, D. L. Andrews, M. Mansuripur, C. Denz, C. Alpmann, P. Banzer, T. Bauer, E. Karimi, L. Marrucci, M. Padgett, M. Ritsch-Marte, N. M. Litchinitser, N. P. Bigelow, C. Rosales-Guzmán, A. Belmonte, J. P. Torres, T. W. Neely, M. Baker, R. Gordon, A. B. Stilgoe, J. Romero, A. G. White, R. Fickler, A. E. Willner, G. Xie, B. McMorran, and A. M. Weiner, “Roadmap on structured light,” J. Opt. 19, 013001 (2017).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

Light Sci. Appl. (1)

H.-L. Zhou, D.-Z. Fu, J.-J. Dong, P. Zhang, D.-X. Chen, X.-L. Cai, F.-L. Li, and X.-L. Zhang, “Orbital angular momentum complex spectrum analyzer for vortex light based on the rotational Doppler effect,” Light Sci. Appl. 6, e16251 (2017).
[Crossref]

Nano Lett. (1)

E. J. R. Vesseur, F. J. García de Abajo, and A. Polman, “Modal decomposition of surface–plasmon whispering gallery resonators,” Nano Lett. 9, 3147–3150 (2009).
[Crossref]

Nat. Commun. (3)

G. Kulkarni, R. Sahu, O. S. Magaña-Loaiza, R. W. Boyd, and A. K. Jha, “Single-shot measurement of the orbital-angular-momentum spectrum of light,” Nat. Commun. 8, 1054 (2017).
[Crossref]

M. Malik, M. Mirhosseini, M. P. Lavery, J. Leach, M. J. Padgett, and R. W. Boyd, “Direct measurement of a 27-dimensional orbital-angular-momentum state vector,” Nat. Commun. 5, 3115 (2014).
[Crossref]

N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, and J. Carpenter, “Laguerre-Gaussian mode sorter,” Nat. Commun. 10, 1865 (2019).
[Crossref]

Nat. Photonics (1)

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order Poincaré sphere beams from a laser,” Nat. Photonics 10, 327–332 (2016).
[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. (2)

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).
[Crossref]

C. Schulze, A. Dudley, D. Flamm, M. Duparré, and A. Forbes, “Measurement of the orbital angular momentum density of light by modal decomposition,” New J. Phys. 15, 073025 (2013).
[Crossref]

Opt. Eng. (3)

G. P. Andersen, L. Dussan, F. Ghebremichael, and K. Chen, “Holographic wavefront sensor,” Opt. Eng. 48, 085801 (2009).
[Crossref]

S. Pachava, R. Dharmavarapu, A. Vijayakumar, S. Jayakumar, A. Manthalkar, A. Dixit, N. K. Viswanathan, B. Srinivasan, and S. Bhattacharya, “Generation and decomposition of scalar and vector modes carrying orbital angular momentum: a review,” Opt. Eng. 59, 041205 (2019).
[Crossref]

S. Scholes, R. Kara, J. Pinnell, V. Rodríguez-Fajardo, and A. Forbes, “Structured light with digital micromirror devices: a guide to best practice,” Opt. Eng. 59, 041202 (2019).
[Crossref]

Opt. Express (19)

M. Mirhosseini, O. S. Magaña-Loaiza, C. Chen, B. Rodenburg, M. Malik, and R. W. Boyd, “Rapid generation of light beams carrying orbital angular momentum,” Opt. Express 21, 30196–30203 (2013).
[Crossref]

G. Lazarev, P.-J. Chen, J. Strauss, N. Fontaine, and A. Forbes, “Beyond the display: phase-only liquid crystal on silicon devices and their applications in photonics,” Opt. Express 27, 16206–16249 (2019).
[Crossref]

G. Ruffato, M. Massari, M. Girardi, G. Parisi, M. Zontini, and F. Romanato, “Non-paraxial design and fabrication of a compact OAM sorter in the telecom infrared,” Opt. Express 27, 24123–24134 (2019).
[Crossref]

A. Dudley, T. Mhlanga, M. Lavery, A. McDonald, F. S. Roux, M. Padgett, and A. Forbes, “Efficient sorting of Bessel beams,” Opt. Express 21, 165–171 (2013).
[Crossref]

C. Rosales-Guzmán, N. Bhebhe, and A. Forbes, “Simultaneous generation of multiple vector beams on a single SLM,” Opt. Express 25, 25697–25706 (2017).
[Crossref]

C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20, 19714–19725 (2012).
[Crossref]

F. Bouchard, N. H. Valencia, F. Brandt, R. Fickler, M. Huber, and M. Malik, “Measuring azimuthal and radial modes of photons,” Opt. Express 26, 31925–31941 (2018).
[Crossref]

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).
[Crossref]

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).
[Crossref]

I. A. Litvin, A. Dudley, F. S. Roux, and A. Forbes, “Azimuthal decomposition with digital holograms,” Opt. Express 20, 10996–11004 (2012).
[Crossref]

A. Trichili, T. Mhlanga, Y. Ismail, F. S. Roux, M. McLaren, M. Zghal, and A. Forbes, “Detection of Bessel beams with digital axicons,” Opt. Express 22, 17553–17560 (2014).
[Crossref]

M. Paurisse, M. Hanna, F. Druon, P. Georges, C. Bellanger, A. Brignon, and J. Huignard, “Phase and amplitude control of a multimode LMA fiber beam by use of digital holography,” Opt. Express 17, 13000–13008 (2009).
[Crossref]

M. Paurisse, L. Lévèque, M. Hanna, F. Druon, and P. Georges, “Complete measurement of fiber modal content by wavefront analysis,” Opt. Express 20, 4074–4084 (2012).
[Crossref]

C. Schulze, A. Lorenz, D. Flamm, A. Hartung, S. Schröter, H. Bartelt, and M. Duparré, “Mode resolved bend loss in few-mode optical fibers,” Opt. Express 21, 3170–3181 (2013).
[Crossref]

B. R. Boruah and M. A. Neil, “Susceptibility to and correction of azimuthal aberrations in singular light beams,” Opt. Express 14, 10377–10385 (2006).
[Crossref]

O. A. Schmidt, C. Schulze, D. Flamm, R. Brüning, T. Kaiser, S. Schröter, and M. Duparré, “Real-time determination of laser beam quality by modal decomposition,” Opt. Express 19, 6741–6748 (2011).
[Crossref]

T. W. Clark, R. F. Offer, S. Franke-Arnold, A. S. Arnold, and N. Radwell, “Comparison of beam generation techniques using a phase only spatial light modulator,” Opt. Express 24, 6249–6264 (2016).
[Crossref]

C. Schulze, S. Ngcobo, M. Duparré, and A. Forbes, “Modal decomposition without a priori scale information,” Opt. Express 20, 27866–27873 (2012).
[Crossref]

A. Jesacher, A. Schwaighofer, S. Fürhapter, C. Maurer, S. Bernet, and M. Ritsch-Marte, “Wavefront correction of spatial light modulators using an optical vortex image,” Opt. Express 15, 5801–5808 (2007).
[Crossref]

Opt. Lett. (9)

F. Stutzki, H. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett. 36, 4572–4574 (2011).
[Crossref]

G. Vallone, “Role of beam waist in Laguerre–Gauss expansion of vortex beams,” Opt. Lett. 42, 1097–1100 (2017).
[Crossref]

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).
[Crossref]

M. A. Bandres and J. C. Gutiérrez-Vega, “Ince–Gaussian beams,” Opt. Lett. 29, 144–146 (2004).
[Crossref]

J. B. Bentley, J. A. Davis, M. A. Bandres, and J. C. Gutiérrez-Vega, “Generation of helical Ince-Gaussian beams with a liquid-crystal display,” Opt. Lett. 31, 649–651 (2006).
[Crossref]

C. Schulze, D. Flamm, M. Duparré, and A. Forbes, “Beam-quality measurements using a spatial light modulator,” Opt. Lett. 37, 4687–4689 (2012).
[Crossref]

D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010).
[Crossref]

Y. Zhou, J. Zhao, Z. Shi, S. M. H. Rafsanjani, M. Mirhosseini, Z. Zhu, A. E. Willner, and R. W. Boyd, “Hermite–Gaussian mode sorter,” Opt. Lett. 43, 5263–5266 (2018).
[Crossref]

B. Ndagano, H. Sroor, M. McLaren, C. Rosales-Guzmán, and A. Forbes, “Beam quality measure for vector beams,” Opt. Lett. 41, 3407–3410 (2016).
[Crossref]

Optica (2)

Phys. Rev. A (3)

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185 (1992).
[Crossref]

M. McLaren, T. Konrad, and A. Forbes, “Measuring the nonseparability of vector vortex beams,” Phys. Rev. A 92, 023833 (2015).
[Crossref]

F. S. Roux and Y. Zhang, “Projective measurements in quantum and classical optical systems,” Phys. Rev. A 90, 033835 (2014).
[Crossref]

Phys. Rev. B (1)

G. Boudarham and M. Kociak, “Modal decompositions of the local electromagnetic density of states and spatially resolved electron energy loss probability in terms of geometric modes,” Phys. Rev. B 85, 245447 (2012).
[Crossref]

Phys. Rev. Lett. (5)

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett. 94, 143902 (2005).
[Crossref]

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).
[Crossref]

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).
[Crossref]

J. Hickmann, E. Fonseca, W. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105, 053904 (2010).
[Crossref]

Proc. SPIE (5)

M. A. Golub, I. N. Sisakian, V. A. Soifer, and G. V. Uvarov, “New measurement techniques for modal power distribution in fibers,” Proc. SPIE 1366, 273–282 (1991).
[Crossref]

M. R. Duparré, V. S. Pavelyev, B. Luedge, E.-B. Kley, V. A. Soifer, and R. M. Kowarschik, “Generation, superposition, and separation of Gauss-Hermite modes by means of does,” Proc. SPIE 3291, 104–114 (1998).
[Crossref]

M. R. Duparré, C. Rockstuhl, A. Letsch, S. Schroeter, and V. S. Pavelyev, “On-line control of laser beam quality by means of diffractive optical components,” Proc. SPIE 4932, 549–559 (2003).
[Crossref]

M. R. Duparré, V. S. Pavelyev, V. A. Soifer, and B. Luedge, “Laser beam characterization by means of diffractive optical correlation filters,” Proc. SPIE 4095, 40–48 (2000).
[Crossref]

M. Duparré, B. Lüdge, and S. Schröter, “On-line characterization of Nd:YAG laser beams by means of modal decomposition using diffractive optical correlation filters,” Proc. SPIE 5962, 755–764 (2005).
[Crossref]

Sci. Rep. (2)

B. Ndagano, I. Nape, B. Perez-Garcia, S. Scholes, R. I. Hernandez-Aranda, T. Konrad, M. P. J. Lavery, and A. Forbes, “A deterministic detector for vector vortex states,” Sci. Rep. 7, 13882 (2017).
[Crossref]

G. Ruffato, M. Girardi, M. Massari, E. Mafakheri, B. Sephton, P. Capaldo, A. Forbes, and F. Romanato, “A compact diffractive sorter for high-resolution demultiplexing of orbital angular momentum beams,” Sci. Rep. 8, 10248 (2018).
[Crossref]

Other (6)

A. E. Siegman, Defining and Measuring Laser Beam Quality (Springer, 1993), pp. 13–28.

A. E. Siegman, “Basic concepts,” in Lasers (University Science Books, 1986), Chap. 2, pp. 7–15.

C. Rosales-Guzmán and A. Forbes, How to Shape Light with Spatial Light Modulators (SPIE, 2017).

V. A. Soifer and M. A. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).

A. Forbes, Laser Beam Propagation: Generation and Propagation of Customized Light (CRC Press, 2014).

https://github.com/JPinnell/Modal-analysis-of-structured-light .

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1.
Fig. 1. Modal decomposition concept. The field on the left-hand side, perhaps unknown in its properties, is resolved into a sum of modes from a known orthonormal basis, shown here as the ${\rm LG}_p^\ell$ modes. The task is to find the unknown coefficients ${c_{\ell ,p}}$. Summing the right-hand side then provides all the information on the left-hand side.
Fig. 2.
Fig. 2. Modal decomposition with an intensity detector. Lenses ${{\rm L}_1}$ and ${{\rm L}_2}$ image the input spatial mode $U({\textbf x})$ onto the SLM encoding $\Phi _n^*({\textbf x})$. The resulting mode is then Fourier transformed with lens ${\rm L_3}$ and detected at the origin of the Fourier plane with a camera or single pixel detector.
Fig. 3.
Fig. 3. ${\rm LG}_{p = 0}^\ell$ decomposition illustrating the effect of the chosen detection cell size for (a) $d = \frac{1}{4}{d_{{\rm min}}}$ (very sensitive to noise), (b) $d \approx {d_{{\rm min}}}$ (accurate), and (c) $d = 3{d_{{\rm min}}}$ (robust but with some crosstalk). Transverse intensity profiles on the diagonal for $\ell = 0$ and $\ell = 5$ are shown above with red disks indicating the detection cell area in each case.
Fig. 4.
Fig. 4. Finding the optical axis: by allowing a large beam to propagate through the detection system, such as a uniform disk (a), a Gaussian fit (b), (c) can be performed at the detection plane (d) to obtain the pixel coordinates $({c_x},{c_y})$ of the optical axis. The size of the detection cell, denoted by the red circle in (d), is then found using Eq. (15) or approximately from the width of the fitted Gaussian.
Fig. 5.
Fig. 5. Modal decomposition with a single-mode fiber. Lenses ${{\rm L}_1}$ and ${{\rm L}_2}$ image the incoming spatial mode $U({\textbf x})$ onto the SLM. The resulting mode is then imaged into the core of the SMF using lenses ${\rm L_3}$ and ${\rm L_4}$. The light from the fiber is coupled to a detector (such as a power meter) that registers a power reading.
Fig. 6.
Fig. 6. Observe how the raw measured spectrum coefficients of a uniform ${\rm LG}_0^\ell$ superposition (a), (b) depend on the choice of basis. In the case of the full LG basis (c), (d), the SLM rescales the amplitudes. In the case of the OAM eigenstates (e), (f), these are not a full basis of the transverse plane.
Fig. 7.
Fig. 7. ${\rm LG}_0^\ell$ decomposition illustrating the consequences of detection (a) $\frac{1}{4}{z_R}$ before the Fourier plane, (b) in the Fourier plane, and (c) $\frac{1}{2}{z_R}$ after the Fourier plane. Transverse intensity profiles (left) at the corresponding plane for a subset of $\ell$ are shown with phase profile insets of each mode in the top-right corner.
Fig. 8.
Fig. 8. ${\rm LG}_0^\ell$ decomposition illustrating the consequence of displacement of the filter’s central axis from that of the input beam in the (a) negative horizontal and (b) positive diagonal directions for 20% of the beam waist. Bar graphs show the decomposition weights along the diagonals (${\ell _{{\rm input}}} = {\ell _{\rm{filter}}}$). Transverse intensity profiles are shown as insets for the subset $\ell = [- 1,1]$ (lower left corner) as well as $\langle \ell = - 10|\ell = - 10\rangle$ (left) and $\langle \ell = 10|\ell = 10\rangle$ (right) (top right corner). Red cross-hairs show the optical axis of the system.
Fig. 9.
Fig. 9. Radial mode decomposition of a ${\rm LG}_{p = 2}^{\ell = 1}$ beam in the ${\rm LG}_{\ell = 1}^p$ basis over $p = [0,4]$ for a match filter mode size ${w_M}$ (a) equal to and (b) at 60% of the input mode size ${w_0}$ with (i) transverse phase and intensity profiles, (ii) intensity distributions in the Fourier plane, and (iii) mode weights.
Fig. 10.
Fig. 10. Setup to perform wavefront correction (a): this is the same as the detection setup in Fig. 2, but the beam should overfill the SLM screen, and the wavefronts should be approximately planar. By encoding only a helical phase (b) on the SLM and using the image of the distorted vortex at the Fourier plane (c), a phase retrieval algorithm can determine the phase map (d) that produces such a beam. The wavefront distortions (e) are isolated by subtracting the ideal phase (b) from the retrieved phase (d). The corrected beam (f) contains very few residual wavefront errors.
Fig. 11.
Fig. 11. Optical overlaps showing the reference mode and the three optical inner-products required to determine ${c_{{p\ell}}}$ in the case where $p = \ell = 1$. Here, ${\cal M}$ denotes the square root of the maximum measured intensity in the detection cell of all the optical overlaps performed.
Fig. 12.
Fig. 12. (a) Full set of experimentally determined expansion coefficients (bars) for each mode in the predetermined set. Red lines indicate the measured amplitude spectrum before correction. With the mode weightings in hand, one can reconstruct the (b) intensity and (c) phase of the field.
Fig. 13.
Fig. 13. (a) Example of a typical quantum experiment where photons from an entanglement source (ES) are decomposed in the OAM eigenstates using a SLM with the projected photons coupled to single-mode fibers and avalanche photodiodes (APD). Detected correlations for different OAM modes when the encoded mode has a Gaussian argument with (b) $\gamma = 0.4$ and (c) $\gamma = 0.9$.
Fig. 14.
Fig. 14. Theoretical ($T$) and experimental ($E$) mode distribution for detected modes using an optical fiber and a SLM as a detector. The $\gamma$ parameter controls the size of the mode on the SLM relative to the fiber mode size.
Fig. 15.
Fig. 15. Breadboard connection diagram for a photodiode and transimpedance amplifier (with variable gain set by a potentiometer), connected to an Arduino Nano via the green wire. The opamp choice is flexible, such as a TL071A.

Tables (2)

Tables Icon

Table 1 Listing 1. Arduino code to continuously read from ADC pin A0 and write the value to the serial port. In addition, the built-in LED is dimmed proportional to the input reading for easy alignment.

Tables Icon

Table 2 Listing 2. A MATLAB script to read 1000 values from the serial port, normalize them, and add them to a vector.

Equations (30)

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

v = n c n e n ,
e m e n = e m | e n = δ m , n ,
c n = e n | v .
U ( x ) = n c n Φ n ( x ) ,
d 2 x Φ m ( x ) Φ n ( x ) = Φ m | Φ n = δ m , n ,
c n = Φ n | U = d 2 x Φ n ( x ) U ( x ) .
U ( x ) SLM + lens W ( k ) = C S d 2 x Φ n ( x ) U ( x ) × exp ( i 2 π λ f k x ) ,
W ( 0 ) = C S d 2 x Φ n ( x ) U ( x ) c n .
| W ( 0 ) | 2 | c n | 2 = ρ n 2 .
( ρ n cos ) 2 = | 1 2 Φ 0 + Φ n | U | 2 ,
( ρ n sin ) 2 = | 1 2 Φ 0 + i Φ n | U | 2 .
( ρ n cos ) 2 = 1 2 ( ρ 0 2 + ρ n 2 + 2 ρ 0 ρ n cos ϕ n ) ,
( ρ n sin ) 2 = 1 2 ( ρ 0 2 + ρ n 2 + 2 ρ 0 ρ n sin ϕ n ) .
ϕ n = a r c t a n ( 2 ( ρ n sin ) 2 ρ 0 2 ρ n 2 2 ( ρ n cos ) 2 ρ 0 2 ρ n 2 ) .
d m i n λ f D ,
c ~ n = d 2 x Φ n ( x ) U ( x ) G ( x ) ,
G ( x ) = 2 π w 2 exp ( r 2 w 2 ) ,
N m a x = V 2 2 ,
V = 2 π λ a N A = 2 π λ a n c o r e 2 n c l a d 2 ,
P n = η | c n | 2 S ( x ) d x ,
P n = η | c n | 2 | G ( x ) | 2 d x = η | c n | 2 .
c n = γ n c ¯ n .
W ( 0 ) S d 2 x Φ n ( x ) M n U ( x ) ,
= 1 M n c n ,
U ( r , ϕ ) = c A ( r ) exp ( i ϕ ) ,
W ( 0 ) c 0 r d r A ( r ) π π d ϕ exp [ i ( n ) ϕ ] , c n 0 r d r A n ( r ) .
U ( x ) = n c n Φ n ( x ) = n c ¯ n Φ ¯ n ( x ) ,
c ¯ n = m c m Φ ¯ n | Φ m .
U = 1 N ( 1 2 e i π L G 0 1 + 1 4 e i π / 4 L G 1 1 + e i π / 2 L G 2 1 + 1 2 L G 0 0 + 1 4 e i π / 4 L G 1 1 + e i π / 2 L G 2 1 ) ,
c ; γ = N ~ ( γ , w , w p ) Φ 1 ( r , ϕ , γ w ) Φ 2 ( r , ϕ ; γ w ) × exp ( r 2 w p 2 ) exp ( 2 r 2 w 2 ) d 2 r .

Metrics

Select as filters


Select Topics Cancel
© Copyright 2022 | Optica Publishing Group. All Rights Reserved