Abstract

Optical transmission through a spiral phase plate is analyzed by treating the device as a Fabry–Perot etalon with an azimuthally varying thickness. The transmitted beam is calculated to contain a coherent superposition of optical vortices with different winding numbers. This yields an intensity profile with a periodic modulation as a function of azimuthal angle where the orientation rotates as a function of the laser frequency. These effects are quantified experimentally and theoretically.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [CrossRef]
  4. S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
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  5. S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B 4, S7–S16 (2002).
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  9. S. Franke-Arnold and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
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  10. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
    [CrossRef]
  11. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
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  12. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
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  13. M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97, 170406 (2006).
    [CrossRef]
  14. R. Kanamoto, E. M. Wright, and P. Meystre, “Quantum dynamics of Raman-coupled Bose–Einstein condensates using Laguerre–Gaussian beams,” Phys. Rev. A 75, 063623 (2007).
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  15. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express 15, 8619–8625 (2007).
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  16. T. P. Simula, N. Nygaard, S. X. Hu, L. A. Collins, B. I. Schneider, and K. Mølmer, “Angular momentum exchange between coherent light and matter fields,” Phys. Rev. A 77, 015401 (2008).
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  24. Z. Dutton and J. Ruostekoski, “Transfer and storage of vortex states in light and matter waves,” Phys. Rev. Lett. 93, 193602 (2004).
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  25. K. T. Kapale and J. P. Dowling, “Vortex phase qubit: generating arbitrary, counter-rotating, coherent superpositions in Bose–Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95, 173601 (2005).
    [CrossRef]
  26. S. Thanvanthri, K. T. Kapale, and J. P. Dowling, “Arbitrary coherent superpositions of quantized vortices in Bose–Einstein condensates via orbital angular momentum of light,” Phys. Rev. A 77, 053825 (2008).
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  29. V. Yu. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
    [CrossRef]
  30. N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
    [CrossRef]
  31. N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
    [CrossRef]
  32. S. van Enk and G. Nienhuis, “Eigenfunction description of laser beams and orbital angular momentum of light,” Opt. Commun. 94, 147–158 (1992).
    [CrossRef]
  33. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
  34. M. J. Padgett and L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B 4, S17–S19 (2002).
    [CrossRef]
  35. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96, 163905 (2006).
    [CrossRef]
  36. F. B. de Colstoun, G. Khitrova, A. V. Fedorov, T. R. Nelson, C. Lowry, T. M. Brennan, B. G. Hammons, and P. D. Maker, “Transverse modes, vortices and vertical-cavity surface-emitting Lasers,” Chaos, Solitons & Fractals 4, 1575–1596 (1994).
  37. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phase plate,” Opt. Commun. 112, 321–327 (1994).
    [CrossRef]
  38. S. S. R. Oemrawsingh, J. A. W. van Houwelingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosterboer, and G. W. ’t Hooft, “Production and characterization of spiral phase plates for optical wavelengths,” Appl. Opt. 43, 688–694 (2004).
    [CrossRef]
  39. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
    [CrossRef]
  40. P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 13101 (2012).
    [CrossRef]
  41. A. E. Siegman, Lasers (University Science Books, 1986).
  42. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  43. M. A. Rob, “Limitation of a wedge etalon for high resolution linewidth measurements,” Opt. Lett. 15, 604–606 (1990).
    [CrossRef]
  44. M. Dobosz and M. Kozuchowski, “Frequency stabilization of a diode by means of an optical wedge,” Meas. Sci. Technol. 23, 035202 (2012).
    [CrossRef]
  45. J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2005).
  46. The angle between the two surfaces of the spiral phase plate depends on radial position: tan(θ(r))=Δh/2πr. The surfaces are approximately parallel, i.e., θ(r)≪1, for r≫Δh, where Δh∼O(λ). For r≤O(λ), fabrication limitations often cause the actual thickness profile of the spiral phase plate to deviate from the ideal profile in Eq. (1). Accordingly, θ(r)≪1 over the same area where Eq. (1) accurately represents the surface of the device.
  47. S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. 6, 103 (2004).
    [CrossRef]
  48. E. Yao, S. Franke-Arnold, J. Courtial, S. Barnett, and M. Padgett, “Fourier relationship between angular position and optical orbital angular momentum,” Opt. Express 14, 9071–9076 (2006).
    [CrossRef]
  49. B. Jack, M. J. Padgett, and S. Franke-Arnold, “Angular diffraction,” New J. Phys. 10, 103013 (2008).
    [CrossRef]
  50. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268(2004).
    [CrossRef]
  51. J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
    [CrossRef]
  52. J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008).
    [CrossRef]
  53. Spiral phase plates are purchased from RPC Photonics, www.rpcphotonics.com. The refractive index is calculated from the dispersion equation of the material used to manufacture the spiral phase plate, and its uncertainty is calculated from the wavelength dependence of the dispersion equation. The spiral step height with uncertainty are experimentally measured values by RPC Photonics. The thickness of the glass substrate and its uncertainty is provided by RPC Photonics. The coefficient of thermal expansion and the thermal coefficient of the refractive index is also provided by RPC Photonics.
  54. Y. S. Rumala, “A new etalon geometry: the spiral phase plate etalon,” Ph.D. thesis (University of Michigan, 2012).

2012 (2)

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 13101 (2012).
[CrossRef]

M. Dobosz and M. Kozuchowski, “Frequency stabilization of a diode by means of an optical wedge,” Meas. Sci. Technol. 23, 035202 (2012).
[CrossRef]

2011 (2)

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[CrossRef]

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5, 343–348 (2011).
[CrossRef]

2009 (1)

K. C. Wright, L. S. Leslie, A. Hansen, and N. P. Bigelow, “Sculpting the vortex state of a spinor BEC,” Phys. Rev. Lett. 102, 030405 (2009).
[CrossRef]

2008 (6)

T. P. Simula, N. Nygaard, S. X. Hu, L. A. Collins, B. I. Schneider, and K. Mølmer, “Angular momentum exchange between coherent light and matter fields,” Phys. Rev. A 77, 015401 (2008).
[CrossRef]

K. C. Wright, L. S. Leslie, and N. P. Bigelow, “Optical control of the internal and external angular momentum of a Bose–Einstein condensate,” Phys. Rev. A 77, 041601 (2008).
[CrossRef]

S. Franke-Arnold and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2, 299–313 (2008).
[CrossRef]

S. Thanvanthri, K. T. Kapale, and J. P. Dowling, “Arbitrary coherent superpositions of quantized vortices in Bose–Einstein condensates via orbital angular momentum of light,” Phys. Rev. A 77, 053825 (2008).
[CrossRef]

B. Jack, M. J. Padgett, and S. Franke-Arnold, “Angular diffraction,” New J. Phys. 10, 103013 (2008).
[CrossRef]

J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008).
[CrossRef]

2007 (4)

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
[CrossRef]

R. Kanamoto, E. M. Wright, and P. Meystre, “Quantum dynamics of Raman-coupled Bose–Einstein condensates using Laguerre–Gaussian beams,” Phys. Rev. A 75, 063623 (2007).
[CrossRef]

S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express 15, 8619–8625 (2007).
[CrossRef]

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

2006 (3)

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef]

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

E. Yao, S. Franke-Arnold, J. Courtial, S. Barnett, and M. Padgett, “Fourier relationship between angular position and optical orbital angular momentum,” Opt. Express 14, 9071–9076 (2006).
[CrossRef]

2005 (3)

K. T. Kapale and J. P. Dowling, “Vortex phase qubit: generating arbitrary, counter-rotating, coherent superpositions in Bose–Einstein condensates via optical angular momentum beams,” Phys. Rev. Lett. 95, 173601 (2005).
[CrossRef]

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, “Nonlocality of high-dimensional two-photon orbital angular momentum states,” Phys. Rev. A 72, 052114 (2005).
[CrossRef]

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

2004 (6)

Z. Dutton and J. Ruostekoski, “Transfer and storage of vortex states in light and matter waves,” Phys. Rev. Lett. 93, 193602 (2004).
[CrossRef]

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[CrossRef]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef]

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268(2004).
[CrossRef]

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. 6, 103 (2004).
[CrossRef]

S. S. R. Oemrawsingh, J. A. W. van Houwelingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosterboer, and G. W. ’t Hooft, “Production and characterization of spiral phase plates for optical wavelengths,” Appl. Opt. 43, 688–694 (2004).
[CrossRef]

2003 (2)

J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. 90, 133901 (2003).
[CrossRef]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef]

2002 (2)

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B 4, S7–S16 (2002).
[CrossRef]

M. J. Padgett and L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B 4, S17–S19 (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]

1994 (3)

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
[CrossRef]

F. B. de Colstoun, G. Khitrova, A. V. Fedorov, T. R. Nelson, C. Lowry, T. M. Brennan, B. G. Hammons, and P. D. Maker, “Transverse modes, vortices and vertical-cavity surface-emitting Lasers,” Chaos, Solitons & Fractals 4, 1575–1596 (1994).

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

1993 (1)

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

1992 (5)

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

N. R. Heckenberg, R. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M. J. Wegener, “Laser beams with phase singularities,” Opt. Quantum Electron. 24, S951–S962 (1992).
[CrossRef]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef]

S. van Enk and G. Nienhuis, “Eigenfunction description of laser beams and orbital angular momentum of light,” Opt. Commun. 94, 147–158 (1992).
[CrossRef]

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

1990 (2)

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

M. A. Rob, “Limitation of a wedge etalon for high resolution linewidth measurements,” Opt. Lett. 15, 604–606 (1990).
[CrossRef]

1983 (1)

1936 (1)

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

1935 (1)

R. A. Beth, “Direct detection of the angular momentum of light,” Phys. Rev. 48, 471 (1935).
[CrossRef]

’t Hooft, G. W.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

S. S. R. Oemrawsingh, J. A. W. van Houwelingen, E. R. Eliel, J. P. Woerdman, E. J. K. Verstegen, J. G. Kloosterboer, and G. W. ’t Hooft, “Production and characterization of spiral phase plates for optical wavelengths,” Appl. Opt. 43, 688–694 (2004).
[CrossRef]

Aiello, A.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, “Nonlocality of high-dimensional two-photon orbital angular momentum states,” Phys. Rev. A 72, 052114 (2005).
[CrossRef]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef]

Aieta, F.

P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 13101 (2012).
[CrossRef]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[CrossRef]

Allen, L.

M. J. Padgett and L. Allen, “Orbital angular momentum exchange in cylindrical-lens mode converters,” J. Opt. B 4, S17–S19 (2002).
[CrossRef]

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

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

Andersen, M. F.

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef]

Arnold, A. S.

Barnett, S.

Barnett, S. M.

J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008).
[CrossRef]

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
[CrossRef]

S. Franke-Arnold, S. M. Barnett, E. Yao, J. Leach, J. Courtial, and M. Padgett, “Uncertainty principle for angular position and angular momentum,” New J. Phys. 6, 103 (2004).
[CrossRef]

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B 4, S7–S16 (2002).
[CrossRef]

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110, 670–678 (1994).
[CrossRef]

Bazhenov, V. Yu.

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

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

Beijersbergen, M. W.

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

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

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

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268(2004).
[CrossRef]

Beth, R. A.

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

R. A. Beth, “Direct detection of the angular momentum of light,” Phys. Rev. 48, 471 (1935).
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P. Genevet, N. Yu, F. Aieta, J. Lin, M. A. Kats, R. Blanchard, M. O. Scully, Z. Gaburro, and F. Capasso, “Ultra-thin plasmonic optical vortex plate based on phase discontinuities,” Appl. Phys. Lett. 100, 13101 (2012).
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B. Jack, M. J. Padgett, and S. Franke-Arnold, “Angular diffraction,” New J. Phys. 10, 103013 (2008).
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K. C. Wright, L. S. Leslie, and N. P. Bigelow, “Optical control of the internal and external angular momentum of a Bose–Einstein condensate,” Phys. Rev. A 77, 041601 (2008).
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S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
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K. C. Wright, L. S. Leslie, A. Hansen, and N. P. Bigelow, “Sculpting the vortex state of a spinor BEC,” Phys. Rev. Lett. 102, 030405 (2009).
[CrossRef]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef]

M. F. Andersen, C. Ryu, P. Cladé, V. Natarajan, A. Vaziri, K. Helmerson, and W. D. Phillips, “Quantized rotation of atoms from photons with orbital angular momentum,” Phys. Rev. Lett. 97, 170406 (2006).
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Science (1)

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[CrossRef]

Other (8)

A. E. Siegman, Lasers (University Science Books, 1986).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed.(Roberts & Company, 2005).

The angle between the two surfaces of the spiral phase plate depends on radial position: tan(θ(r))=Δh/2πr. The surfaces are approximately parallel, i.e., θ(r)≪1, for r≫Δh, where Δh∼O(λ). For r≤O(λ), fabrication limitations often cause the actual thickness profile of the spiral phase plate to deviate from the ideal profile in Eq. (1). Accordingly, θ(r)≪1 over the same area where Eq. (1) accurately represents the surface of the device.

Spiral phase plates are purchased from RPC Photonics, www.rpcphotonics.com. The refractive index is calculated from the dispersion equation of the material used to manufacture the spiral phase plate, and its uncertainty is calculated from the wavelength dependence of the dispersion equation. The spiral step height with uncertainty are experimentally measured values by RPC Photonics. The thickness of the glass substrate and its uncertainty is provided by RPC Photonics. The coefficient of thermal expansion and the thermal coefficient of the refractive index is also provided by RPC Photonics.

Y. S. Rumala, “A new etalon geometry: the spiral phase plate etalon,” Ph.D. thesis (University of Michigan, 2012).

D. L. Andrews, ed., Structured Light and Its Applications: An Introduction to Phase-Structured Beams and Nanoscale Optical Forces (Elsevier, 2008).

J. P. Torres and L. Torner, Twisted Photons: Applications of Light with Orbital Angular Momentum (Wiley-VCH, 2011).

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Figures (5)

Fig. 1.
Fig. 1.

Multiple-beam interference. Approximating a spiral phase plate as a Fabry–Perot etalon with an azimuthally varying thickness h(ϕ) [Eq. (1)] yields reflected, uR=u1+u3+u5+, and transmitted, uT=u2+u4+u6+, waves that contain a coherent superposition of optical vortices with different winding numbers.

Fig. 2.
Fig. 2.

Measured optical transmission through a spiral phase plate. Intensity profiles (a) I(r,ϕ,0)e2r2/ω02 and (b) I(r,ϕ,h0+Δh)e2r2/ω02|t(ϕ)|2 at the input and output planes of the device, respectively. The dark horizontal line in (b) is due to the sudden change in material thickness at ϕ={0,2π} (Fig. 1). (c) Transmittance, |t(ϕ)|2, as a function of azimuthal angle, ϕ. Each black dot represents an angular wedge, and the dashed curve is a weighted fit of the data to Eq. (4) with fit parameters r2fit=0.2067±0.0004 and βfit=5.012±0.003. The data affected by the dark horizontal line in (b) are plotted as open circles and excluded from the fit. The measured central value and uncertainty of each parameter is obtained from their respective histograms containing the fit parameter of approximately 300 interference patterns (Appendix A). The uncertainty is the 68% confidence interval of the histogram.

Fig. 3.
Fig. 3.

Measured rotation of the optical interference pattern at each frequency for two data sets, with rotation rates (a) (Δϕ/Δν)=(73±3)mrad/GHz (dots) and (b) (Δϕ/Δν)=(113±2)mrad/GHz (circles). Each data point represents the single shot measurement of the orientation of the optical interference pattern as determined from a weighted fit of the data at each frequency. The size of the error bar is on the order the size of each data point where the vertical error bar is in the fit routine 68% confidence level, e.g., (1/β)(ϕ0±δϕ0)=1.439±0.005rad, νν0±δ(νν0)=11.254±0.009GHz. The horizontal error bar on the laser frequency is on order 9 MHz. The uncertainty on the rotation rates is determined from a linear fit of the experimental data. The theory rotation rate in Eq. (7) is comparable to the experimental slopes of (a) and (b) at the level of 3% and 50%, respectively.

Fig. 4.
Fig. 4.

Distribution of approximately 300 fit values obtained from fitting the experimentally measured optical interference pattern to Eq. (4) in order to extract the central value and uncertainty (68% confidence interval of distribution): (a) r2fit=0.2067±0.0004 and (b) βfit=5.012±0.003. Histogram of fit parameters show an approximate Gaussian distribution (dashed curve).

Fig. 5.
Fig. 5.

Experimentally measured modulation depth of the optical angular interference pattern at each laser frequency for two data sets, (a) one laser frequency scan (black dots) and (b) other laser frequency scan (open circles) over 15 GHz, corresponding to the rotation rates in Figs. 3(a) and 3(b), respectively. Each data point represents the single-shot measurement of the amplitude reflectivity, r2fit, of the optical interference pattern as determined from a weighted fit of the data at each laser frequency. The uncertainty on each data point is the 68% confidence interval from the fit routine.

Equations (25)

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

h(ϕ)=h0+Δhϕ2π(0ϕ2π).
tspp(ϕ)=u(r,ϕ,h0+Δh)u(r,ϕ,0)=t0e+iαϕ,
tFP(ϕ)=u(r,ϕ,h0+Δh)u(r,ϕ,0)=u2+u4+u6+,
=t2t1t0e+iαϕm=0r22meim(βϕ+ϕ0),
=t2t1t0e+iαϕ1r22e+i(βϕ+ϕ0),
|tFP(ϕ)|2=11+4|r2|2(1|r2|2)2sin2(βϕ+ϕ02),
p=1N|12π02πtFP(ϕ)eiϕdϕ|2,
=1N|12π02πt2t1t0eiαϕm=0(r22me+iβm(ϕ+ϕ0))eiϕdϕ|2,
=1N(|t2t1t0|2|r2m|2).
pm=(1|r2|4)|r2|4m.
Δϕ=1βdϕ0dνΔν=2πh0ΔhΔνν.
Δϕ(1)=ϵϕ0ΔTNϕ02πβ.
|tFP(ϕ)|2=11+4|r2|2(1|r2|2)2sin2(βϕ+ϕ02).
|tFP(ϕ)|2=11+4|r2(1+ϵr2ΔT)|2(1|r2(1+ϵr2ΔT)|2)2sin2(β(1+ϵβΔT)ϕ+ϕ0(1+ϵϕ0ΔT)2).
ϕ0=Nϕ02π+ϕ0fit.
sin2(βϕ+ϕ0fit2)=0.
ϕ(1)=2π(ϕ0fit+Δϕ0fit)β+Δβ,
=2π(ϕ0fit+Δϕ0fit)β(1+Δββ),
2π(ϕ0fit+Δϕ0fit)β(1Δββ),
=2πϕ0fitβΔϕ0fitβ2πϕ0fitβΔββ+ϕ0fitβΔββ,
=ϕ(1)Δϕ0fitβ(2πϕ0fitΔϕ0fitβ)Δββ,
=ϕ(1)+Δϕ(1)+Δϕ1(1).
Δϕ(1)=Δϕ0fitβ=ϵϕ0ΔTNϕ02πβ.
ΔϕΔν=2πh0Δh1ν.
Δϕ+δϕ2πh0(1+ϵgeΔT)Δh(1+ϵseΔT)Δνν.

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