Abstract

We experimentally demonstrate that ptychographic coherent diffractive imaging can be used to simultaneously characterize the amplitude and phase of bichromatic orbital angular momenta-shaped vortex beams, which consist of a fundamental field, together with its copropagating second-harmonic field. In contrast to most other orbital angular momentum characterization methods, this approach solves for the complex field of a hyperspectral beam. This technique can also be used to characterize other phase-structured illumination beams, and, in the future, will be able to be extended to other complex fields in the extreme ultraviolet or X-ray spectral regions, as well as to matter waves.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (4)

2017 (5)

T. Saito, Y. Takeo, and H. Mimura, “Precise characterization of focused vortex beams,” Jpn. J. Appl. Phys. 56(9), 092501 (2017).
[Crossref]

M. Miranda, M. Kotur, P. Rudawski, C. Guo, A. Harth, A. L’Huillier, and C. L. Arnold, “Spatiotemporal characterization of ultrashort optical vortex pulses,” J. Mod. Opt. 64(sup4), S1–S6 (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(1), 013001 (2017).
[Crossref]

D. F. Gardner, M. Tanksalvala, E. R. Shanblatt, X. Zhang, B. R. Galloway, C. L. Porter, R. Karl, C. Bevis, D. E. Adams, H. C. Kapteyn, M. M. Murnane, and G. F. Mancini, “Subwavelength coherent imaging of periodic samples using a 13.5 nm tabletop high-harmonic light source,” Nat. Photonics 11(4), 259–263 (2017).
[Crossref]

C. L. Porter, M. Tanksalvala, M. Gerrity, G. Miley, X. Zhang, C. Bevis, E. Shanblatt, R. Karl, M. M. Murnane, D. E. Adams, and H. C. Kapteyn, “General-purpose, wide field-of-view reflection imaging with a tabletop 13 nm light source,” Optica 4(12), 1552 (2017).
[Crossref]

2016 (6)

L. Janicijevic and S. Topuzoski, “Gaussian laser beam transformation into an optical vortex beam by helical lens,” J. Mod. Opt. 63(2), 164–176 (2016).
[Crossref]

L. Rego, J. S. Román, A. Picón, L. Plaja, and C. Hernández-García, “Nonperturbative twist in the generation of extreme-ultraviolet vortex beams,” Phys. Rev. Lett. 117(16), 163202 (2016).
[Crossref] [PubMed]

A. Aadhi, P. Vaity, P. Chithrabhanu, S. G. Reddy, S. Prabakar, and R. P. Singh, “Non-coaxial superposition of vector vortex beams,” Appl. Opt. 55(5), 1107–1111 (2016).
[Crossref] [PubMed]

E. R. Shanblatt, C. L. Porter, D. F. Gardner, G. F. Mancini, R. M. Karl, M. D. Tanksalvala, C. S. Bevis, V. H. Vartanian, H. C. Kapteyn, D. E. Adams, and M. M. Murnane, “Quantitative chemically specific coherent diffractive imaging of reactions at buried interfaces with few nanometer precision,” Nano Lett. 16(9), 5444–5450 (2016).
[Crossref] [PubMed]

B. Zhang, D. F. Gardner, M. H. Seaberg, E. R. Shanblatt, C. L. Porter, R. Karl, C. A. Mancuso, H. C. Kapteyn, M. M. Murnane, and D. E. Adams, “Ptychographic hyperspectral spectromicroscopy with an extreme ultraviolet high harmonic comb,” Opt. Express 24(16), 18745–18754 (2016).
[Crossref] [PubMed]

S. N. Alperin, R. D. Niederriter, J. T. Gopinath, and M. E. Siemens, “Quantitative measurement of the orbital angular momentum of light with a single, stationary lens,” Opt. Lett. 41(21), 5019–5022 (2016).
[Crossref] [PubMed]

2015 (9)

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
[Crossref]

L. Yan, P. Gregg, E. Karimi, A. Rubano, L. Marrucci, R. Boyd, and S. Ramachandran, “Q-plate enabled spectrally diverse orbital-angular-momentum conversion for stimulated emission depletion microscopy,” Optica 2(10), 900–903 (2015).
[Crossref]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
[Crossref]

O. Kfir, P. Grychtol, E. Turgut, R. Knut, D. Zusin, D. Popmintchev, T. Popmintchev, H. Nembach, J. M. Shaw, A. Fleischer, H. Kapteyn, M. Murnane, and O. Cohen, “Generation of bright phase-matched circularly-polarized extreme ultraviolet high harmonics,” Nat. Photonics 9(2), 99–105 (2015).
[Crossref]

T. Fan, P. Grychtol, R. Knut, C. Hernández-García, D. D. Hickstein, D. Zusin, C. Gentry, F. J. Dollar, C. A. Mancuso, C. W. Hogle, O. Kfir, D. Legut, K. Carva, J. L. Ellis, K. M. Dorney, C. Chen, O. G. Shpyrko, E. E. Fullerton, O. Cohen, P. M. Oppeneer, D. B. Milošević, A. Becker, A. A. Jaroń-Becker, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright circularly polarized soft X-ray high harmonics for X-ray magnetic circular dichroism,” Proc. Natl. Acad. Sci. U.S.A. 112(46), 14206–14211 (2015).
[Crossref] [PubMed]

D. D. Hickstein, F. J. Dollar, P. Grychtol, J. L. Ellis, R. Knut, C. Hernandez-Garcia, D. Zusin, C. Gentry, J. M. Shaw, T. Fan, K. M. Dorney, A. Becker, A. Jaron-Becker, H. C. Kapteyn, M. M. Murnane, and C. G. Durfee, “Non-collinear generation of angularly isolated circularly polarized high harmonics,” Nat. Photonics 9(11), 743–750 (2015).
[Crossref]

A. A. Kovalev and V. V. Kotlyar, “Orbital angular momentum of superposition of identical shifted vortex beams,” J. Opt. Soc. Am. A 32(10), 1805–1810 (2015).
[Crossref] [PubMed]

B. Zhang, D. F. Gardner, M. D. Seaberg, E. R. Shanblatt, H. C. Kapteyn, M. M. Murnane, and D. E. Adams, “High contrast 3D imaging of surfaces near the wavelength limit using tabletop EUV ptychography,” Ultramicroscopy 158, 98–104 (2015).
[Crossref] [PubMed]

C. W. Clark, R. Barankov, M. G. Huber, M. Arif, D. G. Cory, and D. A. Pushin, “Controlling neutron orbital angular momentum,” Nature 525(7570), 504–506 (2015).
[Crossref] [PubMed]

2014 (2)

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
[Crossref] [PubMed]

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
[Crossref] [PubMed]

2013 (5)

J. He, X. Wang, D. Hu, J. Ye, S. Feng, Q. Kan, and Y. Zhang, “Generation and evolution of the terahertz vortex beam,” Opt. Express 21(17), 20230–20239 (2013).
[Crossref] [PubMed]

C. Hernández-García, A. Picón, J. San Román, and L. Plaja, “Attosecond extreme ultraviolet vortices from high-order harmonic generation,” Phys. Rev. Lett. 111(8), 083602 (2013).
[Crossref] [PubMed]

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494(7435), 68–71 (2013).
[Crossref] [PubMed]

M. Sato, T. Higuchi, N. Kanda, K. Konishi, K. Yoshioka, T. Suzuki, K. Misawa, and M. Kuwata-Gonokami, “Terahertz polarization pulse shaping with arbitrary field control,” Nat. Photonics 7(9), 724–731 (2013).
[Crossref]

J. Bahrdt, K. Holldack, P. Kuske, R. Müller, M. Scheer, and P. Schmid, “First observation of photons carrying orbital angular momentum in undulator radiation,” Phys. Rev. Lett. 111(3), 034801 (2013).
[Crossref] [PubMed]

2011 (2)

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
[Crossref] [PubMed]

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3(2), 161 (2011).
[Crossref]

2010 (1)

J. M. Hickmann, E. J. S. Fonseca, W. C. 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(5), 053904 (2010).
[Crossref] [PubMed]

2009 (2)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Chapter 5 Singular optics: Optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
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A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109(10), 1256–1262 (2009).
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2008 (1)

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108(5), 481–487 (2008).
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2006 (2)

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
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F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
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2004 (1)

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 4795–4797 (2004).
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2002 (1)

2001 (1)

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature 412(6844), 313–316 (2001).
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1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
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1998 (1)

A. Rundquist, C. G. Durfee, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft x-rays,” Science 280(5368), 1412–1415 (1998).
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1997 (2)

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56(5), 4064–4075 (1997).
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J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56(5), 4193–4196 (1997).
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1996 (2)

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54(5), R3742–R3745 (1996).
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G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127(4-6), 183–188 (1996).
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1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. 75(5), 826–829 (1995).
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1994 (2)

M. Harris, C. A. Hill, and J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106(4-6), 161–166 (1994).
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M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112(5-6), 321–327 (1994).
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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(11), 8185–8189 (1992).
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1990 (1)

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).

1952 (1)

D. Sayre, “Some implications of a theorem due to Shannon,” Acta Crystallogr. 5(6), 843 (1952).
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1936 (1)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115–125 (1936).
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Aadhi, A.

Adams, D. E.

C. L. Porter, M. Tanksalvala, M. Gerrity, G. Miley, X. Zhang, C. Bevis, E. Shanblatt, R. Karl, M. M. Murnane, D. E. Adams, and H. C. Kapteyn, “General-purpose, wide field-of-view reflection imaging with a tabletop 13 nm light source,” Optica 4(12), 1552 (2017).
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D. F. Gardner, M. Tanksalvala, E. R. Shanblatt, X. Zhang, B. R. Galloway, C. L. Porter, R. Karl, C. Bevis, D. E. Adams, H. C. Kapteyn, M. M. Murnane, and G. F. Mancini, “Subwavelength coherent imaging of periodic samples using a 13.5 nm tabletop high-harmonic light source,” Nat. Photonics 11(4), 259–263 (2017).
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E. R. Shanblatt, C. L. Porter, D. F. Gardner, G. F. Mancini, R. M. Karl, M. D. Tanksalvala, C. S. Bevis, V. H. Vartanian, H. C. Kapteyn, D. E. Adams, and M. M. Murnane, “Quantitative chemically specific coherent diffractive imaging of reactions at buried interfaces with few nanometer precision,” Nano Lett. 16(9), 5444–5450 (2016).
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B. Zhang, D. F. Gardner, M. H. Seaberg, E. R. Shanblatt, C. L. Porter, R. Karl, C. A. Mancuso, H. C. Kapteyn, M. M. Murnane, and D. E. Adams, “Ptychographic hyperspectral spectromicroscopy with an extreme ultraviolet high harmonic comb,” Opt. Express 24(16), 18745–18754 (2016).
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B. Zhang, D. F. Gardner, M. D. Seaberg, E. R. Shanblatt, H. C. Kapteyn, M. M. Murnane, and D. E. Adams, “High contrast 3D imaging of surfaces near the wavelength limit using tabletop EUV ptychography,” Ultramicroscopy 158, 98–104 (2015).
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Agrawal, A.

B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
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Ahmed, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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Allen, L.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes,” Phys. Rev. A 56(5), 4193–4196 (1997).
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K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54(5), R3742–R3745 (1996).
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G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre-Gaussian modes at millimetre-wave frequencies by use of a spiral phaseplate,” Opt. Commun. 127(4-6), 183–188 (1996).
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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(11), 8185–8189 (1992).
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Alperin, S. N.

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(1), 013001 (2017).
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B. J. McMorran, A. Agrawal, I. M. Anderson, A. A. Herzing, H. J. Lezec, J. J. McClelland, and J. Unguris, “Electron vortex beams with high quanta of orbital angular momentum,” Science 331(6014), 192–195 (2011).
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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(1), 013001 (2017).
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Anzolin, G.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
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Arif, M.

C. W. Clark, R. Barankov, M. G. Huber, M. Arif, D. G. Cory, and D. A. Pushin, “Controlling neutron orbital angular momentum,” Nature 525(7570), 504–506 (2015).
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Arnold, C. L.

M. Miranda, M. Kotur, P. Rudawski, C. Guo, A. Harth, A. L’Huillier, and C. L. Arnold, “Spatiotemporal characterization of ultrashort optical vortex pulses,” J. Mod. Opt. 64(sup4), S1–S6 (2017).
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A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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Ashrafi, S.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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Backus, S.

A. Rundquist, C. G. Durfee, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft x-rays,” Science 280(5368), 1412–1415 (1998).
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Bahrdt, J.

J. Bahrdt, K. Holldack, P. Kuske, R. Müller, M. Scheer, and P. Schmid, “First observation of photons carrying orbital angular momentum in undulator radiation,” Phys. Rev. Lett. 111(3), 034801 (2013).
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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(1), 013001 (2017).
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Balcou, P.

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(1), 013001 (2017).
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Bao, C.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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Barankov, R.

C. W. Clark, R. Barankov, M. G. Huber, M. Arif, D. G. Cory, and D. A. Pushin, “Controlling neutron orbital angular momentum,” Nature 525(7570), 504–506 (2015).
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Barbieri, C.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
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Batey, D. J.

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
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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(1), 013001 (2017).
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Baynard, E.

Bazhenov, V. Y.

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser beams with screw dislocations in their wavefronts,” JETP Lett. 52, 429–431 (1990).

Becker, A.

D. D. Hickstein, F. J. Dollar, P. Grychtol, J. L. Ellis, R. Knut, C. Hernandez-Garcia, D. Zusin, C. Gentry, J. M. Shaw, T. Fan, K. M. Dorney, A. Becker, A. Jaron-Becker, H. C. Kapteyn, M. M. Murnane, and C. G. Durfee, “Non-collinear generation of angularly isolated circularly polarized high harmonics,” Nat. Photonics 9(11), 743–750 (2015).
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T. Fan, P. Grychtol, R. Knut, C. Hernández-García, D. D. Hickstein, D. Zusin, C. Gentry, F. J. Dollar, C. A. Mancuso, C. W. Hogle, O. Kfir, D. Legut, K. Carva, J. L. Ellis, K. M. Dorney, C. Chen, O. G. Shpyrko, E. E. Fullerton, O. Cohen, P. M. Oppeneer, D. B. Milošević, A. Becker, A. A. Jaroń-Becker, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright circularly polarized soft X-ray high harmonics for X-ray magnetic circular dichroism,” Proc. Natl. Acad. Sci. U.S.A. 112(46), 14206–14211 (2015).
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Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112(5-6), 321–327 (1994).
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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(11), 8185–8189 (1992).
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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(1), 013001 (2017).
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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(1), 013001 (2017).
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R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115–125 (1936).
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Bevis, C.

D. F. Gardner, M. Tanksalvala, E. R. Shanblatt, X. Zhang, B. R. Galloway, C. L. Porter, R. Karl, C. Bevis, D. E. Adams, H. C. Kapteyn, M. M. Murnane, and G. F. Mancini, “Subwavelength coherent imaging of periodic samples using a 13.5 nm tabletop high-harmonic light source,” Nat. Photonics 11(4), 259–263 (2017).
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C. L. Porter, M. Tanksalvala, M. Gerrity, G. Miley, X. Zhang, C. Bevis, E. Shanblatt, R. Karl, M. M. Murnane, D. E. Adams, and H. C. Kapteyn, “General-purpose, wide field-of-view reflection imaging with a tabletop 13 nm light source,” Optica 4(12), 1552 (2017).
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Bevis, C. S.

E. R. Shanblatt, C. L. Porter, D. F. Gardner, G. F. Mancini, R. M. Karl, M. D. Tanksalvala, C. S. Bevis, V. H. Vartanian, H. C. Kapteyn, D. E. Adams, and M. M. Murnane, “Quantitative chemically specific coherent diffractive imaging of reactions at buried interfaces with few nanometer precision,” Nano Lett. 16(9), 5444–5450 (2016).
[Crossref] [PubMed]

Bianchini, A.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[Crossref] [PubMed]

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(1), 013001 (2017).
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Bliokh, K. Y.

K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015).
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Boyd, R.

Boyd, R. W.

G. Gariepy, J. Leach, K. T. Kim, T. J. Hammond, E. Frumker, R. W. Boyd, and P. B. Corkum, “Creating high-harmonic beams with controlled orbital angular momentum,” Phys. Rev. Lett. 113(15), 153901 (2014).
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Bunk, O.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108(5), 481–487 (2008).
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Cao, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, “Optical communications using orbital angular momentum beams,” Adv. Opt. Photonics 7(1), 66–106 (2015).
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Carva, K.

T. Fan, P. Grychtol, R. Knut, C. Hernández-García, D. D. Hickstein, D. Zusin, C. Gentry, F. J. Dollar, C. A. Mancuso, C. W. Hogle, O. Kfir, D. Legut, K. Carva, J. L. Ellis, K. M. Dorney, C. Chen, O. G. Shpyrko, E. E. Fullerton, O. Cohen, P. M. Oppeneer, D. B. Milošević, A. Becker, A. A. Jaroń-Becker, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright circularly polarized soft X-ray high harmonics for X-ray magnetic circular dichroism,” Proc. Natl. Acad. Sci. U.S.A. 112(46), 14206–14211 (2015).
[Crossref] [PubMed]

Chang, Z.

A. Rundquist, C. G. Durfee, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft x-rays,” Science 280(5368), 1412–1415 (1998).
[Crossref] [PubMed]

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
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J. M. Hickmann, E. J. S. Fonseca, W. C. 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(5), 053904 (2010).
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T. Fan, P. Grychtol, R. Knut, C. Hernández-García, D. D. Hickstein, D. Zusin, C. Gentry, F. J. Dollar, C. A. Mancuso, C. W. Hogle, O. Kfir, D. Legut, K. Carva, J. L. Ellis, K. M. Dorney, C. Chen, O. G. Shpyrko, E. E. Fullerton, O. Cohen, P. M. Oppeneer, D. B. Milošević, A. Becker, A. A. Jaroń-Becker, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright circularly polarized soft X-ray high harmonics for X-ray magnetic circular dichroism,” Proc. Natl. Acad. Sci. U.S.A. 112(46), 14206–14211 (2015).
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Chithrabhanu, P.

Clark, C. W.

C. W. Clark, R. Barankov, M. G. Huber, M. Arif, D. G. Cory, and D. A. Pushin, “Controlling neutron orbital angular momentum,” Nature 525(7570), 504–506 (2015).
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D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
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D. F. Gardner, M. Tanksalvala, E. R. Shanblatt, X. Zhang, B. R. Galloway, C. L. Porter, R. Karl, C. Bevis, D. E. Adams, H. C. Kapteyn, M. M. Murnane, and G. F. Mancini, “Subwavelength coherent imaging of periodic samples using a 13.5 nm tabletop high-harmonic light source,” Nat. Photonics 11(4), 259–263 (2017).
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B. Zhang, D. F. Gardner, M. D. Seaberg, E. R. Shanblatt, H. C. Kapteyn, M. M. Murnane, and D. E. Adams, “High contrast 3D imaging of surfaces near the wavelength limit using tabletop EUV ptychography,” Ultramicroscopy 158, 98–104 (2015).
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D. D. Hickstein, F. J. Dollar, P. Grychtol, J. L. Ellis, R. Knut, C. Hernandez-Garcia, D. Zusin, C. Gentry, J. M. Shaw, T. Fan, K. M. Dorney, A. Becker, A. Jaron-Becker, H. C. Kapteyn, M. M. Murnane, and C. G. Durfee, “Non-collinear generation of angularly isolated circularly polarized high harmonics,” Nat. Photonics 9(11), 743–750 (2015).
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T. Fan, P. Grychtol, R. Knut, C. Hernández-García, D. D. Hickstein, D. Zusin, C. Gentry, F. J. Dollar, C. A. Mancuso, C. W. Hogle, O. Kfir, D. Legut, K. Carva, J. L. Ellis, K. M. Dorney, C. Chen, O. G. Shpyrko, E. E. Fullerton, O. Cohen, P. M. Oppeneer, D. B. Milošević, A. Becker, A. A. Jaroń-Becker, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Bright circularly polarized soft X-ray high harmonics for X-ray magnetic circular dichroism,” Proc. Natl. Acad. Sci. U.S.A. 112(46), 14206–14211 (2015).
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C. L. Porter, M. Tanksalvala, M. Gerrity, G. Miley, X. Zhang, C. Bevis, E. Shanblatt, R. Karl, M. M. Murnane, D. E. Adams, and H. C. Kapteyn, “General-purpose, wide field-of-view reflection imaging with a tabletop 13 nm light source,” Optica 4(12), 1552 (2017).
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E. R. Shanblatt, C. L. Porter, D. F. Gardner, G. F. Mancini, R. M. Karl, M. D. Tanksalvala, C. S. Bevis, V. H. Vartanian, H. C. Kapteyn, D. E. Adams, and M. M. Murnane, “Quantitative chemically specific coherent diffractive imaging of reactions at buried interfaces with few nanometer precision,” Nano Lett. 16(9), 5444–5450 (2016).
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C. L. Porter, M. Tanksalvala, M. Gerrity, G. Miley, X. Zhang, C. Bevis, E. Shanblatt, R. Karl, M. M. Murnane, D. E. Adams, and H. C. Kapteyn, “General-purpose, wide field-of-view reflection imaging with a tabletop 13 nm light source,” Optica 4(12), 1552 (2017).
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D. F. Gardner, M. Tanksalvala, E. R. Shanblatt, X. Zhang, B. R. Galloway, C. L. Porter, R. Karl, C. Bevis, D. E. Adams, H. C. Kapteyn, M. M. Murnane, and G. F. Mancini, “Subwavelength coherent imaging of periodic samples using a 13.5 nm tabletop high-harmonic light source,” Nat. Photonics 11(4), 259–263 (2017).
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Figures (5)

Fig. 1
Fig. 1 Experimental schematic for simultaneously imaging an orbital angular momentum beam and its second harmonic. A laser beam with center wavelength 790 nm propagates through a spiral phase plate (SPP) of = 2 and is focused to a BBO crystal using a lens (L1). The fundamental and second harmonic beams both illuminate a resolution test target, and are Fourier transformed via another lens (L2). The target is moved with respect to the two beams, and diffraction patterns are collected on a CMOS camera. Theoretically rendered helical wavefronts of red (790 nm) and blue (395 nm) OAM beams are shown at the top. M1 and M2: mirrors.
Fig. 2
Fig. 2 Reconstructed test target from the experimental data (first column), simulated transverse field at the target (second column), reconstructed transverse field at the target from the experimental data (third column) and sagittal cross section of the reconstructed fields, propagated close to the focus (fourth column) for a bichromatic field (first and second row, �� = 790 nm and �� = 395 nm respectively) and a single-color field (bottom row, �� = 532 nm). In all cases, the electric field amplitude (arbitrary units) is represented in brightness while the phase is represented as a hue, as indicated by the color wheel in (j). In the second column, the transverse fields were simulated by applying the spiral phase plate and the focusing lens phase to an input TEM00 Gaussian beam with adjustable radius and phase curvature, and then propagating the field to the location of the target.
Fig. 3
Fig. 3 Retrieved OAM spectrum of the two reconstructed vortex beam fields obtained by Fourier transformation of the fields along the azimuthal direction for different radii, for (a) 790 nm and (b) 395 nm. The color map corresponds to the spectrum in arbitrary units (a.u.). Note that the resolution in the OAM number is limited to one from Fourier transform of the azimuthal coordinate; the plot has been interpolated for smoothness. In addition, the Fourier transform is in the range  = ± 60, but here we present the section of the spectrum with the highest yield. High yield is seen at   = 2 for �� = 790 nm and at   = 4 for �� = 395 nm as expected.
Fig. 4
Fig. 4 Extracting the spiral phase plate (SPP) structure from the reconstructed fundamental field. (a) 3D illustration of the SPP geometry, with 32 discrete steps that are applied in two consecutive ramps. (b) Optical microscope image at the center of the SPP. The reconstructed fundamental field from ptychography is numerically propagated to the plane of the SPP, and is shown in (c) amplitude (arbitrary units) and (d) phase. The white lines in (c) are shown as a guide to the eye. The amplitude of the field was radially integrated and is shown in (e). After large period oscillations have been removed for visibility, the number of peaks agrees with the number of discrete steps in the SPP used in the experiment.
Fig. 5
Fig. 5 A constant phase plot of the 3D numerically propagated beams from the experimental reconstructions. (a) shows the fundamental (left, red) and the second harmonic (right, blue) beams individually. In (b) and (c), the fundamental and the harmonic beams are combined with different relative phase shifts (left) to produce contrasting total fields (right, magenta), assuming that two fields have exactly the same amplitude. The arrow in (a) indicates the propagation direction.

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