Abstract

In this paper we study both theoretically and experimentally a method to characterize the amplitude and phase of a paraxial optical beam. The method is based on the spiral phase interferometry technique, recently proposed. We theoretically analyze how to adapt the original proposal to deal with the special characteristics of finite optical beams. Finally, we compare a series of numerical and experimental results to show the advantages and limitations of our proposal.

© 2008 Optical Society of America

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References

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  1. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, 1991)
    [CrossRef]
  2. M. S. Soskin and M. V. Vasnetsov, "Singular optics," in Progress in Optics, E. Wolf, ed., (North-Holand, Amstredam, 2001), Vol. 42, pp. 219-276.
  3. P. Galajda and P. Ormos, "Complex micromachines produced and driven by light," Appl. Phys. Lett. 78, 249-251 (2001).
    [CrossRef]
  4. K. Ladavac and D. G. Grier, "Microoptomechanical pumps assembled and driven by holographic optical vortex arrays," Opt. Express 12, 1144-1149 (2004).
    [CrossRef] [PubMed]
  5. R. J. Voogd, M. Singh, S. Pereira, A. van de Nes, and J. Braat, "The use of orbital angular momentum of light beams for super-high density optical data storage," OSA Annual Meeting, Rochester, NY, October 2004, paper FTuG14.
  6. L. Torner, J. P. Torres, and S. Carrasco, "Digital spiral imaging," Opt. Express 13, 873 (2005).
    [CrossRef] [PubMed]
  7. G. Molina-Terriza, L. Rebane, J. P. Torres, L. Torner and S. Carrasco, "Probing canonical geometrical objects by digital spiral imaging," J. Europ. Opt. Soc. Rap. Public. 2, 07014 (2007).
    [CrossRef]
  8. G. Swartzlander, "The optical vortex lens," Opt. Photonics News 17, 39-43 (2006).
    [CrossRef]
  9. L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," in Progress in Optics, E.Wolf, ed., (North-Holand, Amstredam, 1999), Vol. 39, pp. 291-372.
  10. G. Molina-Terriza, J. P. Torres J. P. and L. Torner, "Management of the orbital angular momentum of light: preparation of photons in multidimensional vector states of angular momentum," Phys. Rev. Lett. 88, 013601 (2002).
    [CrossRef] [PubMed]
  11. H. He, M. E. J. Friese, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorbing particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
    [CrossRef] [PubMed]
  12. J. W. R. Tabosa and D. V. Petrov, "Optical pumping of orbital angular momentum of light in cold cesium atoms," Phys. Rev. Lett. 83, 4967-4970 (1999).
    [CrossRef]
  13. M. F. Andersen,  et al., "Quantized rotation of atoms from photons with orbital angular momentum," Phys. Rev. Lett. 97, 170406 (2006).
    [CrossRef] [PubMed]
  14. R. Inoue,  et al., "Entanglement of orbital angular momentum states between an ensemble of cold atoms and a photon," Phys. Rev. A 74, 053809 (2006).
    [CrossRef]
  15. G. Molina-Terriza, J. P. Torres and L. Torner, "Twisted photons," Nat. Physics 3, 305-310 (2007).
    [CrossRef]
  16. B. J. Smith, B. Killett, M. G. Raymer, I. A. Walmsley, and K. Banaszek, "Measurement of the transverse spatial quantum state of light at the single-photon level," Opt. Lett. 30, 3365-3367 (2005).
    [CrossRef]
  17. S. F¨urhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, "Spiral interferometry," Opt. Lett. 30, 1953-1955 (2005).
    [CrossRef] [PubMed]
  18. A. Jesacher, S. F¨urhapter, S. Bernet, and M. Ritsch-Marte, "Spiral interferogram analysis," J. Opt. Soc. Am. A 23, 1400-1409 (2006).
    [CrossRef]
  19. S. F¨urhapter, A. Jesacher, S. Bernet, and M. Ritsch-Marte, "Spiral phase contrast imaging in microscopy," Opt. Express 13, 689-694 (2005).
    [CrossRef] [PubMed]
  20. A. Jesacher, S. F¨urhapter, S. Bernet, and M. Ritsch-Marte, "Shadow effects in spiral phase contrast microscopy," Phys. Rev. Lett. 94, 233902 (2005).
    [CrossRef] [PubMed]
  21. S. Bernet, A. Jesacher, S. F¨urhapter, C. Maurer, and M. Ritsch-Marte, "Quantitative imaging of complex samples by spiral phase contrast microscopy," Opt. Express 14, 3792-3805 (2006).
    [CrossRef] [PubMed]
  22. N. R. Heckenberg, R. G. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M.J. Wegener, "Laser Beams with Phase Singularities," Opt. Quantum Electron. 24, S951-S962 (1992).
    [CrossRef]
  23. A. Vaziri, G. Weihs, and A. Zeilinger, "Superpositions of the orbital angular momentum for applications in quantum experiments," J. Opt. B: Quantum Semiclass. Opt. 4, 47-51 (2002).
    [CrossRef]
  24. G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradil, and A. Zeilinger, "Triggered Qutrits for Quantum Communication Protocols," Phys. Rev. Lett. 92, 168903 (2004).
    [CrossRef]

2007

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

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

2006

M. F. Andersen,  et al., "Quantized rotation of atoms from photons with orbital angular momentum," Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef] [PubMed]

R. Inoue,  et al., "Entanglement of orbital angular momentum states between an ensemble of cold atoms and a photon," Phys. Rev. A 74, 053809 (2006).
[CrossRef]

G. Swartzlander, "The optical vortex lens," Opt. Photonics News 17, 39-43 (2006).
[CrossRef]

S. Bernet, A. Jesacher, S. F¨urhapter, C. Maurer, and M. Ritsch-Marte, "Quantitative imaging of complex samples by spiral phase contrast microscopy," Opt. Express 14, 3792-3805 (2006).
[CrossRef] [PubMed]

A. Jesacher, S. F¨urhapter, S. Bernet, and M. Ritsch-Marte, "Spiral interferogram analysis," J. Opt. Soc. Am. A 23, 1400-1409 (2006).
[CrossRef]

2005

2004

G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradil, and A. Zeilinger, "Triggered Qutrits for Quantum Communication Protocols," Phys. Rev. Lett. 92, 168903 (2004).
[CrossRef]

K. Ladavac and D. G. Grier, "Microoptomechanical pumps assembled and driven by holographic optical vortex arrays," Opt. Express 12, 1144-1149 (2004).
[CrossRef] [PubMed]

2002

A. Vaziri, G. Weihs, and A. Zeilinger, "Superpositions of the orbital angular momentum for applications in quantum experiments," J. Opt. B: Quantum Semiclass. Opt. 4, 47-51 (2002).
[CrossRef]

G. Molina-Terriza, J. P. Torres J. P. and L. Torner, "Management of the orbital angular momentum of light: preparation of photons in multidimensional vector states of angular momentum," Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

2001

P. Galajda and P. Ormos, "Complex micromachines produced and driven by light," Appl. Phys. Lett. 78, 249-251 (2001).
[CrossRef]

1999

J. W. R. Tabosa and D. V. Petrov, "Optical pumping of orbital angular momentum of light in cold cesium atoms," Phys. Rev. Lett. 83, 4967-4970 (1999).
[CrossRef]

1995

H. He, M. E. J. Friese, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorbing particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

1992

N. R. Heckenberg, R. G. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M.J. Wegener, "Laser Beams with Phase Singularities," Opt. Quantum Electron. 24, S951-S962 (1992).
[CrossRef]

Andersen, M. F.

M. F. Andersen,  et al., "Quantized rotation of atoms from photons with orbital angular momentum," Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef] [PubMed]

Banaszek, K.

Bernet, S.

Carrasco, S.

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

L. Torner, J. P. Torres, and S. Carrasco, "Digital spiral imaging," Opt. Express 13, 873 (2005).
[CrossRef] [PubMed]

F¨urhapter, S.

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorbing particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Galajda, P.

P. Galajda and P. Ormos, "Complex micromachines produced and driven by light," Appl. Phys. Lett. 78, 249-251 (2001).
[CrossRef]

Grier, D. G.

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorbing particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorbing particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

N. R. Heckenberg, R. G. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M.J. Wegener, "Laser Beams with Phase Singularities," Opt. Quantum Electron. 24, S951-S962 (1992).
[CrossRef]

Hradil, Z.

G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradil, and A. Zeilinger, "Triggered Qutrits for Quantum Communication Protocols," Phys. Rev. Lett. 92, 168903 (2004).
[CrossRef]

Inoue, R.

R. Inoue,  et al., "Entanglement of orbital angular momentum states between an ensemble of cold atoms and a photon," Phys. Rev. A 74, 053809 (2006).
[CrossRef]

Jesacher, A.

Killett, B.

Ladavac, K.

Maurer, C.

McDuff, R. G.

N. R. Heckenberg, R. G. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M.J. Wegener, "Laser Beams with Phase Singularities," Opt. Quantum Electron. 24, S951-S962 (1992).
[CrossRef]

Molina-Terriza, G.

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

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

G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradil, and A. Zeilinger, "Triggered Qutrits for Quantum Communication Protocols," Phys. Rev. Lett. 92, 168903 (2004).
[CrossRef]

G. Molina-Terriza, J. P. Torres J. P. and L. Torner, "Management of the orbital angular momentum of light: preparation of photons in multidimensional vector states of angular momentum," Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

Ormos, P.

P. Galajda and P. Ormos, "Complex micromachines produced and driven by light," Appl. Phys. Lett. 78, 249-251 (2001).
[CrossRef]

Petrov, D. V.

J. W. R. Tabosa and D. V. Petrov, "Optical pumping of orbital angular momentum of light in cold cesium atoms," Phys. Rev. Lett. 83, 4967-4970 (1999).
[CrossRef]

Raymer, M. G.

Rebane, L.

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

Rehacek, J.

G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradil, and A. Zeilinger, "Triggered Qutrits for Quantum Communication Protocols," Phys. Rev. Lett. 92, 168903 (2004).
[CrossRef]

Ritsch-Marte, M.

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorbing particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

N. R. Heckenberg, R. G. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M.J. Wegener, "Laser Beams with Phase Singularities," Opt. Quantum Electron. 24, S951-S962 (1992).
[CrossRef]

Smith, B. J.

Smith, C. P.

N. R. Heckenberg, R. G. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M.J. Wegener, "Laser Beams with Phase Singularities," Opt. Quantum Electron. 24, S951-S962 (1992).
[CrossRef]

Swartzlander, G.

G. Swartzlander, "The optical vortex lens," Opt. Photonics News 17, 39-43 (2006).
[CrossRef]

Tabosa, J. W. R.

J. W. R. Tabosa and D. V. Petrov, "Optical pumping of orbital angular momentum of light in cold cesium atoms," Phys. Rev. Lett. 83, 4967-4970 (1999).
[CrossRef]

Torner, L.

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

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

L. Torner, J. P. Torres, and S. Carrasco, "Digital spiral imaging," Opt. Express 13, 873 (2005).
[CrossRef] [PubMed]

Torres, J. P.

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

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

L. Torner, J. P. Torres, and S. Carrasco, "Digital spiral imaging," Opt. Express 13, 873 (2005).
[CrossRef] [PubMed]

Vaziri, A.

G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradil, and A. Zeilinger, "Triggered Qutrits for Quantum Communication Protocols," Phys. Rev. Lett. 92, 168903 (2004).
[CrossRef]

A. Vaziri, G. Weihs, and A. Zeilinger, "Superpositions of the orbital angular momentum for applications in quantum experiments," J. Opt. B: Quantum Semiclass. Opt. 4, 47-51 (2002).
[CrossRef]

Walmsley, I. A.

Wegener, M.J.

N. R. Heckenberg, R. G. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M.J. Wegener, "Laser Beams with Phase Singularities," Opt. Quantum Electron. 24, S951-S962 (1992).
[CrossRef]

Weihs, G.

A. Vaziri, G. Weihs, and A. Zeilinger, "Superpositions of the orbital angular momentum for applications in quantum experiments," J. Opt. B: Quantum Semiclass. Opt. 4, 47-51 (2002).
[CrossRef]

Zeilinger, A.

G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradil, and A. Zeilinger, "Triggered Qutrits for Quantum Communication Protocols," Phys. Rev. Lett. 92, 168903 (2004).
[CrossRef]

A. Vaziri, G. Weihs, and A. Zeilinger, "Superpositions of the orbital angular momentum for applications in quantum experiments," J. Opt. B: Quantum Semiclass. Opt. 4, 47-51 (2002).
[CrossRef]

Appl. Phys. Lett.

P. Galajda and P. Ormos, "Complex micromachines produced and driven by light," Appl. Phys. Lett. 78, 249-251 (2001).
[CrossRef]

J. Europ. Opt. Soc. Rap. Public.

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

J. Opt. B: Quantum Semiclass. Opt.

A. Vaziri, G. Weihs, and A. Zeilinger, "Superpositions of the orbital angular momentum for applications in quantum experiments," J. Opt. B: Quantum Semiclass. Opt. 4, 47-51 (2002).
[CrossRef]

J. Opt. Soc. Am. A

Nat. Physics

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

Opt. Express

Opt. Lett.

Opt. Photonics News

G. Swartzlander, "The optical vortex lens," Opt. Photonics News 17, 39-43 (2006).
[CrossRef]

Opt. Quantum Electron.

N. R. Heckenberg, R. G. McDuff, C. P. Smith, H. Rubinsztein-Dunlop, and M.J. Wegener, "Laser Beams with Phase Singularities," Opt. Quantum Electron. 24, S951-S962 (1992).
[CrossRef]

Phys. Rev. A

R. Inoue,  et al., "Entanglement of orbital angular momentum states between an ensemble of cold atoms and a photon," Phys. Rev. A 74, 053809 (2006).
[CrossRef]

Phys. Rev. Lett.

A. Jesacher, S. F¨urhapter, S. Bernet, and M. Ritsch-Marte, "Shadow effects in spiral phase contrast microscopy," Phys. Rev. Lett. 94, 233902 (2005).
[CrossRef] [PubMed]

G. Molina-Terriza, J. P. Torres J. P. and L. Torner, "Management of the orbital angular momentum of light: preparation of photons in multidimensional vector states of angular momentum," Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

H. He, M. E. J. Friese, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorbing particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

J. W. R. Tabosa and D. V. Petrov, "Optical pumping of orbital angular momentum of light in cold cesium atoms," Phys. Rev. Lett. 83, 4967-4970 (1999).
[CrossRef]

M. F. Andersen,  et al., "Quantized rotation of atoms from photons with orbital angular momentum," Phys. Rev. Lett. 97, 170406 (2006).
[CrossRef] [PubMed]

G. Molina-Terriza, A. Vaziri, J. Rehacek, Z. Hradil, and A. Zeilinger, "Triggered Qutrits for Quantum Communication Protocols," Phys. Rev. Lett. 92, 168903 (2004).
[CrossRef]

Other

R. J. Voogd, M. Singh, S. Pereira, A. van de Nes, and J. Braat, "The use of orbital angular momentum of light beams for super-high density optical data storage," OSA Annual Meeting, Rochester, NY, October 2004, paper FTuG14.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (John Wiley & Sons, 1991)
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, "Singular optics," in Progress in Optics, E. Wolf, ed., (North-Holand, Amstredam, 2001), Vol. 42, pp. 219-276.

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," in Progress in Optics, E.Wolf, ed., (North-Holand, Amstredam, 1999), Vol. 39, pp. 291-372.

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Figures (5)

Fig. 1.
Fig. 1.

Block diagram of our system. (a) represents the optical processing, OFT mean optical Fourier transform, and (b) represents the numerical post-processing to recover the field, FFT and IFFT are the fast Fourier transform and its inverse, and sqrt{} is the square root operation. More details are given in the text.

Fig. 2.
Fig. 2.

Numerical example of the reconstruction system. (a) Intensity of the input field. (b) Phase of the input field. (c) Amplitude of the Fourier transform of the input beam. The white dot indicates the position of the center of the filter. (d) One of the filters used, the white dot indicates the center of the filter. (e), (f), (g) Output intensities of the system, corresponding to the different filters used. (h) Intensity of the recovered field. (i) Phase of the recovered field.

Fig. 3.
Fig. 3.

Sketch of the experimental setup. A computer generated phase hologram is illuminated with a collimated diode laser light to produce a Laguerre-Gaussian-like beam in the object plane, using the lens L1 and an iris (to select the first order of diffraction). Then L2 makes the Fourier transform of the object and puts it on the SLM surface, where the filters are displayed. After the filtering, we make the Fourier transform again with L3 and we rescale the image to fit the CCD chip with the imaging system 1. With the imaging system 2 we make an image of the SLM on the CCD to find a proper point where to center the filters.

Fig. 4.
Fig. 4.

Characterization of a Gaussian beam with four embedded phase singularities. Upper row, simulation of the expected field. Lower row, experimental reconstruction. (a) and (c) Intensity pattern with the four zeroes associated with the four phase dislocations. (b) and (d) Phase pattern.

Fig. 5.
Fig. 5.

Characterization of a Gaussian beam with a phase jump. Upper row, simulation of the expected field using a 0.8π phase jump. Lower row, experimental reconstruction. (a) and (c) Intensity pattern. (b) and (d) Phase pattern.

Equations (10)

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

H k ( x , y ) = { exp ( i θ ( x , y ) ) if x 2 + y 2 > R 2 e i α k if x 2 + y 2 R 2
I out ( k ) ( x out , y out ) { { E in x in y in } H k ( x , y ) } 2
= A x out y out + exp ( i α k ) B x out y out 2
A r out θ out = R r dr 0 2 π E ~ in ( r , θ ) exp ( i θ ) exp ( ir out r cos ( θ out θ ) ) d θ
B r out θ out = 0 R r dr 0 2 π E ~ in ( r , θ ) exp ( ir out r cos ( θ out θ ) ) d θ .
I c = 1 3 k = 1 3 I out ( k ) e i α k = A * ( x out , y out ) B ( x out , y out )
I tot = 1 3 k = 1 3 I out ( k ) = A ( x out , y out ) 2 + B ( x out , y out ) 2
      Φ = arg { { { I c ( x out , y out ) } e i arctan y x } } ( x R x + y R y )
E rec = I out 0 exp ( i Φ )
B ( x , y ) J 1 ( R ( x δ x ) 2 + ( y δ y ) 2 ) R ( x δ x ) 2 + ( y δ y ) 2 exp ( i ( x R x + y R y ) )

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