Abstract

Single-pixel cameras reconstruct images from a stream of spatial projection measurements recorded with a single-element detector, which itself has no spatial resolution. This enables the creation of imaging systems that can take advantage of the ultra-fast response times of single-element detectors. Here we present a single-pixel camera with a temporal resolution of 200 ps in the visible and short-wave infrared wavelengths, used here to study the transit time of distinct spatial modes transmitted through few-mode and orbital angular momentum mode conserving optical fiber. Our technique represents a way to study the spatial and temporal characteristics of light propagation in multimode optical fibers, which may find use in optical fiber design and communications.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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    [Crossref] [PubMed]

2018 (4)

L. Olivieri, J. S. Totero Gongora, A. Pasquazi, and M. Peccianti, “Time-resolved nonlinear ghost imaging,” ACS Photonics 5, 3379–3388 (2018).
[Crossref]

D. Faccio and A. Velten, “A trillion frames per second: the techniques and applications of light-in-flight photography,” Rep. Prog. Phys. 81, 105901 (2018).
[Crossref] [PubMed]

D. E. B. Flaes, J. Stopka, S. Turtaev, J. F. de Boer, T. Tyc, and T. Čižmár, “Robustness of light-transport processes to bending deformations in graded-index multimode waveguides,” Phys. Rev. Lett. 120, 233901 (2018).
[Crossref]

X. Ren, P. W. R. Connolly, A. Halimi, Y. Altmann, S. McLaughlin, I. Gyongy, R. K. Henderson, and G. S. Buller, “High-resolution depth profiling using a range-gated CMOS SPAD quanta image sensor,” Opt. Express 26, 5541–5557 (2018).
[Crossref] [PubMed]

2017 (4)

2016 (4)

M.-J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

R. I. Stantchev, B. Sun, S. M. Hornett, P. A. Hobson, G. M. Gibson, M. J. Padgett, and E. Hendry, “Noninvasive, near-field terahertz imaging of hidden objects using a single-pixel detector,” Sci. Adv. 2, e1600190 (2016).
[Crossref] [PubMed]

M.-J. Sun, M. P. Edgar, D. B. Phillips, G. M. Gibson, and M. J. Padgett, “Improving the signal-to-noise ratio of single-pixel imaging using digital microscanning,” Opt. Express 24, 10476–10485 (2016).
[Crossref] [PubMed]

P. Gregg, P. Kristensen, and S. Ramachandran, “13.4km OAM state propagation by recirculating fiber loop,” Opt. Express 24, 18938–18947 (2016).
[Crossref] [PubMed]

2015 (4)

P. Gregg, P. Kristensen, and S. Ramachandran, “Conservation of orbital angular momentum in air-core optical fibers,” Optica 2, 267–270 (2015).
[Crossref]

M. Plöschner, T. Tyc, and T. Cizmár, “Seeing through chaos in multimode fibres,” Nat. Photonics 9, 529 (2015).
[Crossref]

M. J. Padgett, F. M. Miatto, M. P. J. Lavery, A. Zeilinger, and R. W. Boyd, “Divergence of an orbital-angular-momentum-carrying beam upon propagation,” New J. Phys. 17, 023011 (2015).
[Crossref]

G. Gariepy, N. Krstajić, R. Henderson, C. Li, R. R. Thomson, G. S. Buller, B. Heshmat, R. Raskar, J. Leach, and D. Faccio, “Single-photon sensitive light-in-fight imaging,” Nat. Commun. 6, 6021 (2015).
[Crossref] [PubMed]

2014 (2)

L. Gao, J. Liang, C. Li, and L. V. Wang, “Single-shot compressed ultrafast photography at one hundred billion frames per second,” Nature 516, 74–77 (2014).
[Crossref] [PubMed]

A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. C. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” Science 343, 58–61 (2014).
[Crossref]

2013 (5)

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

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
[Crossref]

S. Ramachandran and P. Kristensen, “Optical vortices in fiber,” Nanophotonics 2, 455–474 (2013).
[Crossref]

S. S. Welsh, M. P. Edgar, R. Bowman, P. Jonathan, B. Sun, and M. J. Padgett, “Fast full-color computational imaging with single-pixel detectors,” Opt. Express 21, 23068–23074 (2013).
[Crossref] [PubMed]

G. A. Howland, D. J. Lum, M. R. Ware, and J. C. Howell, “Photon counting compressive depth mapping,” Opt. Express 21, 23822–23837 (2013).
[Crossref] [PubMed]

2012 (2)

2011 (2)

2009 (2)

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[Crossref]

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696 (2009).
[Crossref]

2008 (2)

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[Crossref]

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16, 7233–7243 (2008).
[Crossref] [PubMed]

1997 (1)

1992 (1)

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

1979 (1)

R. Olshansky, “Propagation in glass optical waveguides,” Rev. Mod. Phys. 51, 341–367 (1979).
[Crossref]

1969 (1)

W. K. Pratt, J. Kane, and H. C. Andrews, “Hadamard transform image coding,” Proc. IEEE 57, 58–68 (1969).
[Crossref]

Allen, L.

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

Altmann, Y.

X. Ren, P. W. R. Connolly, A. Halimi, Y. Altmann, S. McLaughlin, I. Gyongy, R. K. Henderson, and G. S. Buller, “High-resolution depth profiling using a range-gated CMOS SPAD quanta image sensor,” Opt. Express 26, 5541–5557 (2018).
[Crossref] [PubMed]

R. Warburton, C. Aniculaesei, M. Clerici, Y. Altmann, G. Gariepy, R. McCracken, D. Reid, S. McLaughlin, M. Petrovich, J. Hayes, R. Henderson, D. Faccio, and J. Leach, “Observation of laser pulse propagation in optical fibers with a SPAD camera,”Sci. Rep. 7, 43302 (2017).
[Crossref] [PubMed]

Andrews, H. C.

W. K. Pratt, J. Kane, and H. C. Andrews, “Hadamard transform image coding,” Proc. IEEE 57, 58–68 (1969).
[Crossref]

Aniculaesei, C.

R. Warburton, C. Aniculaesei, M. Clerici, Y. Altmann, G. Gariepy, R. McCracken, D. Reid, S. McLaughlin, M. Petrovich, J. Hayes, R. Henderson, D. Faccio, and J. Leach, “Observation of laser pulse propagation in optical fibers with a SPAD camera,”Sci. Rep. 7, 43302 (2017).
[Crossref] [PubMed]

Bai, N.

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[Crossref]

Barankov, R. A.

Beijersbergen, M. W.

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

Bickham, S.

Bolle, C. A.

Bowman, R.

Boyd, R. W.

M. J. Padgett, F. M. Miatto, M. P. J. Lavery, A. Zeilinger, and R. W. Boyd, “Divergence of an orbital-angular-momentum-carrying beam upon propagation,” New J. Phys. 17, 023011 (2015).
[Crossref]

Bozinovic, N.

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

Bromberg, Y.

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[Crossref]

Buller, G. S.

X. Ren, P. W. R. Connolly, A. Halimi, Y. Altmann, S. McLaughlin, I. Gyongy, R. K. Henderson, and G. S. Buller, “High-resolution depth profiling using a range-gated CMOS SPAD quanta image sensor,” Opt. Express 26, 5541–5557 (2018).
[Crossref] [PubMed]

G. Gariepy, N. Krstajić, R. Henderson, C. Li, R. R. Thomson, G. S. Buller, B. Heshmat, R. Raskar, J. Leach, and D. Faccio, “Single-photon sensitive light-in-fight imaging,” Nat. Commun. 6, 6021 (2015).
[Crossref] [PubMed]

Chen, S.

Chung, K. M.

Cizmár, T.

M. Plöschner, T. Tyc, and T. Cizmár, “Seeing through chaos in multimode fibres,” Nat. Photonics 9, 529 (2015).
[Crossref]

Cižmár, T.

D. E. B. Flaes, J. Stopka, S. Turtaev, J. F. de Boer, T. Tyc, and T. Čižmár, “Robustness of light-transport processes to bending deformations in graded-index multimode waveguides,” Phys. Rev. Lett. 120, 233901 (2018).
[Crossref]

Clerici, M.

R. Warburton, C. Aniculaesei, M. Clerici, Y. Altmann, G. Gariepy, R. McCracken, D. Reid, S. McLaughlin, M. Petrovich, J. Hayes, R. Henderson, D. Faccio, and J. Leach, “Observation of laser pulse propagation in optical fibers with a SPAD camera,”Sci. Rep. 7, 43302 (2017).
[Crossref] [PubMed]

Colaço, A.

A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. C. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” Science 343, 58–61 (2014).
[Crossref]

Connolly, P. W. R.

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[Crossref]

de Boer, J. F.

D. E. B. Flaes, J. Stopka, S. Turtaev, J. F. de Boer, T. Tyc, and T. Čižmár, “Robustness of light-transport processes to bending deformations in graded-index multimode waveguides,” Phys. Rev. Lett. 120, 233901 (2018).
[Crossref]

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[Crossref]

Edgar, M. P.

Essiambre, R.-J.

Faccio, D.

D. Faccio and A. Velten, “A trillion frames per second: the techniques and applications of light-in-flight photography,” Rep. Prog. Phys. 81, 105901 (2018).
[Crossref] [PubMed]

R. Warburton, C. Aniculaesei, M. Clerici, Y. Altmann, G. Gariepy, R. McCracken, D. Reid, S. McLaughlin, M. Petrovich, J. Hayes, R. Henderson, D. Faccio, and J. Leach, “Observation of laser pulse propagation in optical fibers with a SPAD camera,”Sci. Rep. 7, 43302 (2017).
[Crossref] [PubMed]

G. Gariepy, N. Krstajić, R. Henderson, C. Li, R. R. Thomson, G. S. Buller, B. Heshmat, R. Raskar, J. Leach, and D. Faccio, “Single-photon sensitive light-in-fight imaging,” Nat. Commun. 6, 6021 (2015).
[Crossref] [PubMed]

Fini, J. M.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
[Crossref]

Flaes, D. E. B.

D. E. B. Flaes, J. Stopka, S. Turtaev, J. F. de Boer, T. Tyc, and T. Čižmár, “Robustness of light-transport processes to bending deformations in graded-index multimode waveguides,” Phys. Rev. Lett. 120, 233901 (2018).
[Crossref]

Gao, L.

L. Gao, J. Liang, C. Li, and L. V. Wang, “Single-shot compressed ultrafast photography at one hundred billion frames per second,” Nature 516, 74–77 (2014).
[Crossref] [PubMed]

Gariepy, G.

R. Warburton, C. Aniculaesei, M. Clerici, Y. Altmann, G. Gariepy, R. McCracken, D. Reid, S. McLaughlin, M. Petrovich, J. Hayes, R. Henderson, D. Faccio, and J. Leach, “Observation of laser pulse propagation in optical fibers with a SPAD camera,”Sci. Rep. 7, 43302 (2017).
[Crossref] [PubMed]

G. Gariepy, N. Krstajić, R. Henderson, C. Li, R. R. Thomson, G. S. Buller, B. Heshmat, R. Raskar, J. Leach, and D. Faccio, “Single-photon sensitive light-in-fight imaging,” Nat. Commun. 6, 6021 (2015).
[Crossref] [PubMed]

Ghalmi, S.

Ghosh, G.

Gibson, G. M.

M.-J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

R. I. Stantchev, B. Sun, S. M. Hornett, P. A. Hobson, G. M. Gibson, M. J. Padgett, and E. Hendry, “Noninvasive, near-field terahertz imaging of hidden objects using a single-pixel detector,” Sci. Adv. 2, e1600190 (2016).
[Crossref] [PubMed]

M.-J. Sun, M. P. Edgar, D. B. Phillips, G. M. Gibson, and M. J. Padgett, “Improving the signal-to-noise ratio of single-pixel imaging using digital microscanning,” Opt. Express 24, 10476–10485 (2016).
[Crossref] [PubMed]

Gnauck, A. H.

Goyal, V. K.

A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. C. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” Science 343, 58–61 (2014).
[Crossref]

Gregg, P.

Gyongy, I.

Hadfield, R. H.

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696 (2009).
[Crossref]

Halimi, A.

Hayes, J.

R. Warburton, C. Aniculaesei, M. Clerici, Y. Altmann, G. Gariepy, R. McCracken, D. Reid, S. McLaughlin, M. Petrovich, J. Hayes, R. Henderson, D. Faccio, and J. Leach, “Observation of laser pulse propagation in optical fibers with a SPAD camera,”Sci. Rep. 7, 43302 (2017).
[Crossref] [PubMed]

He, Y.

Henderson, R.

R. Warburton, C. Aniculaesei, M. Clerici, Y. Altmann, G. Gariepy, R. McCracken, D. Reid, S. McLaughlin, M. Petrovich, J. Hayes, R. Henderson, D. Faccio, and J. Leach, “Observation of laser pulse propagation in optical fibers with a SPAD camera,”Sci. Rep. 7, 43302 (2017).
[Crossref] [PubMed]

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R. I. Stantchev, D. B. Phillips, P. Hobson, S. M. Hornett, M. J. Padgett, and E. Hendry, “Compressed sensing with near-field THz radiation,” Optica 4, 989–992 (2017).
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R. I. Stantchev, B. Sun, S. M. Hornett, P. A. Hobson, G. M. Gibson, M. J. Padgett, and E. Hendry, “Noninvasive, near-field terahertz imaging of hidden objects using a single-pixel detector,” Sci. Adv. 2, e1600190 (2016).
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Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
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M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
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P. Gregg, P. Kristensen, and S. Ramachandran, “13.4km OAM state propagation by recirculating fiber loop,” Opt. Express 24, 18938–18947 (2016).
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N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
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M.-J. Sun, M. P. Edgar, D. B. Phillips, G. M. Gibson, and M. J. Padgett, “Improving the signal-to-noise ratio of single-pixel imaging using digital microscanning,” Opt. Express 24, 10476–10485 (2016).
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N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
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A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. C. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” Science 343, 58–61 (2014).
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L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
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R. I. Stantchev, D. B. Phillips, P. Hobson, S. M. Hornett, M. J. Padgett, and E. Hendry, “Compressed sensing with near-field THz radiation,” Optica 4, 989–992 (2017).
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M.-J. Sun, M. P. Edgar, D. B. Phillips, G. M. Gibson, and M. J. Padgett, “Improving the signal-to-noise ratio of single-pixel imaging using digital microscanning,” Opt. Express 24, 10476–10485 (2016).
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G. Gariepy, N. Krstajić, R. Henderson, C. Li, R. R. Thomson, G. S. Buller, B. Heshmat, R. Raskar, J. Leach, and D. Faccio, “Single-photon sensitive light-in-fight imaging,” Nat. Commun. 6, 6021 (2015).
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ACS Photonics (1)

L. Olivieri, J. S. Totero Gongora, A. Pasquazi, and M. Peccianti, “Time-resolved nonlinear ghost imaging,” ACS Photonics 5, 3379–3388 (2018).
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Appl. Opt. (2)

IEEE Signal Process. Mag. (1)

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[Crossref]

Nanophotonics (1)

S. Ramachandran and P. Kristensen, “Optical vortices in fiber,” Nanophotonics 2, 455–474 (2013).
[Crossref]

Nat. Commun. (2)

M.-J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

G. Gariepy, N. Krstajić, R. Henderson, C. Li, R. R. Thomson, G. S. Buller, B. Heshmat, R. Raskar, J. Leach, and D. Faccio, “Single-photon sensitive light-in-fight imaging,” Nat. Commun. 6, 6021 (2015).
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Nat. Photonics (3)

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
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M. Plöschner, T. Tyc, and T. Cizmár, “Seeing through chaos in multimode fibres,” Nat. Photonics 9, 529 (2015).
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R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photonics 3, 696 (2009).
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Nature (1)

L. Gao, J. Liang, C. Li, and L. V. Wang, “Single-shot compressed ultrafast photography at one hundred billion frames per second,” Nature 516, 74–77 (2014).
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New J. Phys. (1)

M. J. Padgett, F. M. Miatto, M. P. J. Lavery, A. Zeilinger, and R. W. Boyd, “Divergence of an orbital-angular-momentum-carrying beam upon propagation,” New J. Phys. 17, 023011 (2015).
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Opt. Express (10)

Z. Zhang, X. Wang, G. Zheng, and J. Zhong, “Hadamard single-pixel imaging versus fourier single-pixel imaging,” Opt. Express 25, 19619–19639 (2017).
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P. Gregg, P. Kristensen, and S. Ramachandran, “13.4km OAM state propagation by recirculating fiber loop,” Opt. Express 24, 18938–18947 (2016).
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G. A. Howland, D. J. Lum, M. R. Ware, and J. C. Howell, “Photon counting compressive depth mapping,” Opt. Express 21, 23822–23837 (2013).
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J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16, 7233–7243 (2008).
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Supplementary Material (3)

NameDescription
» Visualization 1       FMF Mode Images
» Visualization 2       FMF Mode Images, Second data set
» Visualization 3       OAM Mode images

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Figures (3)

Fig. 1
Fig. 1 The experimental set-up of our light-in-flight camera system. A pulsed super continuum laser illuminates the input to the optical fiber under test. The illumination wavelength is selected with a bandpass filter within the laser assembly (not shown). Light exiting from the output of the optical fiber is collected by a microscope objective and transmitted to a digital micro-mirror device (DMD), displaying a series of mask patterns. The light reflected from the on mirrors of the DMD is transmitted to the single-pixel detector. The photon arrival time is measured by the time correlated single-photon counting (TCSPC) electronics, timed with respect to an electronic synchronisation signal from the laser controller.

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Fig. 2
Fig. 2 The upper plot shows the total intensity transmitted from the FMF as a function of time, calculated by summing the contributions from all patterns. The normalised 2D intensity distribution at several different times are shown below, with the raw image and the image after kernel smoothing, the line colour corresponds to the time of the image frame shown. The images of the modes have been normalised giving each image the maximum contrast. The full data is shown in Visualization 1.

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Fig. 3
Fig. 3 The upper plot shows the integral intensity transmitted as a function of time from the 200 m length of OAM fiber. The vertical lines show the predicted arrival time of the optical modes. For each mode the measured intensity pattern and the respective modelled intensity of the OAM modes are shown. The unlabelled peaks in the plot are identified as higher order modes that were not included in the mode predictions. The images of the modes have been normalised giving each image the maximum contrast. The full data is shown in Visualization 3.

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Equations (3)

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I 2D ( t ) = ∑ i = 1 N M ( t ) i ⋅ H i ,
K ( x , x ′ , y , y ′ , t , t ′ ) = exp  ( − ( x − x ′ ) 2 2 σ xy 2 ) ⋅ exp  ( − ( y − y ′ ) 2 2 σ xy 2 ) ⋅ exp  ( − ( t − t ′ ) 2 2 σ t 2 )
Y ( x ′ , y ′ , t ′ ) = ∭ K ( x , x ′ , y , y ′ , t , t ′ ) ⋅ X ( x , y , t ) d x d y d t .

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