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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 6 — Mar. 25, 2013
  • pp: 7505–7510
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Polarization holograms allow highly efficient generation of complex light beams

U. Ruiz, P. Pagliusi, C. Provenzano, K. Volke-Sepúlveda, and Gabriella Cipparrone  »View Author Affiliations


Optics Express, Vol. 21, Issue 6, pp. 7505-7510 (2013)
http://dx.doi.org/10.1364/OE.21.007505


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Abstract

We report a viable method to generate complex beams, such as the non-diffracting Bessel and Weber beams, which relies on the encoding of amplitude information, in addition to phase and polarization, using polarization holography. The holograms are recorded in polarization sensitive films by the interference of a reference plane wave with a tailored complex beam, having orthogonal circular polarizations. The high efficiency, the intrinsic achromaticity and the simplicity of use of the polarization holograms make them competitive with respect to existing methods and attractive for several applications. Theoretical analysis, based on the Jones formalism, and experimental results are shown.

© 2013 OSA

1. Introduction

Here, we present a method for the generation of PIOFs, directly from an incident plane wave, by means of PHs. This is the first demonstration of non-quantized amplitude information storage in a PH. The high efficiency predicted for PHs recorded with orthogonal circularly polarized plane waves is exploited here to efficiently generate complex structured beams. To illustrate the method, we report on the generation of BBs and WBs by PHs, characterized by comparable accuracy and significantly higher efficiency with respect to the synthetic phase hologram methods [20

20. V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated Phase Computer Holograms for the Accurate Encoding of Scalar Complex Fields,” J. Opt. Soc. Am. A 24(11), 3500–3507 (2007). [CrossRef] [PubMed]

].

The generation of complex beams by means of PHs is relevant not only as an alternative method, but it may also pave the way to study unexplored aspects about the dynamical properties of light and how they change in interaction with a medium. In addition, the possibility of encoding polarization, phase and amplitude in a single optical element may be important also in quantum information.

2. Theory

We adopt the Jones formalism to describe the generation and the diffraction of the PHs, which are assumed to be recorded in a material with only photoinduced linear optical anisotropy. The aim of this work is to generate complex light fields Ec=A(q1,q2)eiϑ(q1,q2), where A(q1, q2) and ϑ(q1, q2) are the amplitude and the phase variation of the complex beam, and (q1, q2) denote either the circular (r,φ) or the parabolic (ν,ζ) coordinates in the transverse plane. We consider the interference of a plane wave Ep and a tailored complex field Em, having opposite circular polarizations, whose propagation axes form an angle α:
Ep=E02(1i)eiδ,Em=E02f(A)(1i)eiϑeiδ,
(1)
where δπx/Λ, ΛλR/2sin(α/2) is the spatial periodicity of the polarization pattern, λR is the recording wavelength and f (A) is an amplitude function that will be determined so as to produce the desired complex field in the read out process.

Assuming a normalized linearly or circularly polarized plane wave (Einlin=(10)T or Eincirc=1/2(1±i)T) impinging on the PH, we obtain
E±1lin=i2sin(2kdβlinE02f(A))eiϑ(1±i),E±1circ=i2sin(2kdβlinE02f(A))Ψeiϑ(1i),
(4)
where Ψ = 1 (Ψ = 0) for the + 1 order and Ψ = 0 (Ψ = 1) for the −1 order when the incident beam is left-handed (right-handed) circularly polarized.

According to Eq. (4) the first-order ( ± 1) diffracted beams have conjugated phase ϑ and opposite circular polarizations, while exchange of energy between the diffracted beams occurs by varying the ellipticity of the incident plane wave, in which the −1 order reproduces the polarization and phase of the complex field Em.

In order to codify also the amplitude A of the complex beam Ec, we determine f(A) in the recording wave Em, such that sin(2kdβlinE02f(A)) is proportional to A. According to Eq. (4), we assume
sin(2kdβlinE02f(A))=ξA,
(5)
where 0<ξ1 ξ is a uniform parameter, so that f(A)=sin1(ξA)/2kdβlinE02.

The parameter ξ is evaluated by considering the PH created by the interference between two plane waves (i.e.,f(A)=A=1). In this case, in fact, Eq. (5) becomes ξ=sin(2kdβlinE02)=ηp, where ηp=(|E+1|2+|E1|2)/|Ein|2 is the total diffraction efficiency, as reported in [1

1. L. Nikolova and P. S. Ramanujam, Polarization Holography, (Cambridge University Press, 2009).

].

In the general case, the total diffraction efficiency can be calculated from Eq. (4)
ηΩ(|E+1|2+|E1|2)dΩΩ|Ein|2dΩ=ηpΩΩ|A|2dΩ,
(6)
where Ω represents the total area in which the complex beam is defined. In the case of the BB A(r,φ)eiϑ(r,φ)=J1(2πkrr)eiφ, where J1 is the first order Bessel function, kr is the spatial frequency, and the diffraction efficiency calculated by Eq. (6) is ηBessel0.15×ηp. In the case of the third order odd WB A(ν,ζ)eiϑ(ν,ζ)=Po(ν,ζ;3) [11

11. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004). [CrossRef] [PubMed]

], the diffraction efficiency is ηWeber0.10×ηp.

3. Experimental investigation

We have experimentally investigated the generation of BBs and WBs, by means of PHs recorded in an amorphous azo-polymer [21

21. G. Cipparrone, P. Pagliusi, C. Provenzano, and V. P. Shibaev, “Polarization holographic recording in amorphous polymer with photoinduced linear and circular birefringence,” J. Phys. Chem. B 114(27), 8900–8904 (2010). [CrossRef] [PubMed]

]. The latter is confined between two glass plates in a 23µm thick film, prepared by heating the polymer above the glass transition temperature Tg and cooling it down to room temperature. The polymer film is exposed to the polarization patterns obtained by the interference of the plane wave Ep and the complex beam Em in Eq. (1), having orthogonal circular polarization. Both the beams are generated from an Ar+ laser (wavelength λR = 488nm) over a circular area Ω0.05cm2 using a phase spatial light modulator (SLM) (PLUTO, HOLOEYE GmbH), addressed by means of a synthetic phase hologram [20

20. V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated Phase Computer Holograms for the Accurate Encoding of Scalar Complex Fields,” J. Opt. Soc. Am. A 24(11), 3500–3507 (2007). [CrossRef] [PubMed]

]. The recording waves are superposed on the recording medium at a small angle α0.4°, which corresponds to a spatial period Λ = 72μm for the PH. The recording intensity of the plane wave Ep (25 mW/cm2) and the exposure time (60s) have been chosen by maximizing the total diffraction efficiency ηp of the PH recorded by two plane waves, in the range of linear photoresponse of the polymer film (Δnlin=βlinE02). Indeed, this latter condition is necessary to fit the theoretical approach (see Eqs. (3) and (5)). Depending on the maximum photoinduced birefringence of the polymer far from saturation and on its absorption at the recording wavelength λR, we have experimentally found a value for ηp = 0.70, corresponding to the cited intensity and exposure time. A circularly polarized He–Ne laser probe beam (λ = 633nm), whose wavelength is far from the absorption band of the polymer, has been used to reconstruct the BBs and WBs from the PHs. A CCD camera has been used to capture the intensity distribution of the complex beams E1circ diffracted by the PHs and compare them with the corresponding BBs and WBs generated by the SLM.

In Fig. 1
Fig. 1 First order BB. (a) Intensity distribution of the beam generated by the SLM, (b) intensity distribution of the beam reconstructed by the PH, (c) the propagated PH-reconstructed BB at the plane z = 30cm, and (d) 1D intensity profiles of the SLM-generated (black line) and PH-reconstructed (red line) BBs.
we analyse the case of the first order BB, i.e Aeiϑ=J1(2πkrr)eiφ, with kr17cm1. According to the selected parameters of the BB and the optical components employed to reconstruct the BB, the maximum non-diffracting distance is zmax ≅ 1.2m. The intensity profiles of the BBs generated by the SLM and reconstructed by the PH are reported in Fig. 1(a) and 1(b), respectively. A comparison between the two beams is reported in Fig. 1(d), which shows the 1D normalized intensity profiles of the SLM-generated (black) and the PH-reconstructed (red) BBs evaluated along the white line in Fig. 1(a) and 1(b). In Fig. 1(c), we show the propagated PH-reconstructed BB at z = 30cm. In order to evaluate quantitatively the agreement between the intensity modulation of the BBs produced by the SLM and the PH, we use the root-mean squared error, RMSE1ΩΩ(ISLMIPH)2dΩ.

For the case of the BBs the RMSE ≅ 0.08, which demonstrates a good agreement between the BBs generated with the two methods. On the other hand, the measured value of the diffraction efficiency is ηBBexp0.10, which is very close to the theoretical value calculated from Eq. (6) ηBB0.15×0.70=0.105, and about one order of magnitude higher than the efficiency for the SLM-generated BBs.

In Fig. 2
Fig. 2 Third order odd parabolic beam. (a) Intensity distribution of the beam generated by the SLM, (b) intensity distribution of the beam reconstructed by the PH, (c) the propagated PH-reconstructed WB at the plane z = 30cm.
we analyse the case of the third order odd WBs A(ν,ζ)eiϑ(ν,ζ)=Po(ν,ζ;3) [11

11. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004). [CrossRef] [PubMed]

]. In this case, the maximum non-diffracting distance is zmax ≅ 0.55m. The intensity profiles of the SLM-generated and PH-reconstructed beams are reported in Fig. 2(a) and 2(b), respectively. In Fig. 2(c), we show the propagated PH-reconstructed WB at z = 30 cm. The good agreement between the intensity modulation of the WBs produced with the two methods is demonstrated by RMSE ≅ 0.06. The measured value of the diffraction efficiency is ηWBsexp0.07, which is very close to the theoretical value calculated from Eq. (6) ηWBs0.10×0.70=0.07, and about one order of magnitude higher than the efficiency for the SLM-generated WBs.

4. Conclusions

In conclusion, we report a method for generation of arbitrary complex beams employing polarization holograms, which results from the recording of the interference pattern between a plane wave and a tailored complex beam, with orthogonal circular polarizations, on a polarization sensitive photoanisotropic material. Exploiting an amorphous azopolymer characterized by linear photoinduced birefringence as recording medium, we demonstrate the generation of circularly polarized Bessel and Weber non-diffracting beams from a plane wave with efficiencies 0.10 and 0.07, respectively, which are 70% of the maximum theoretical diffraction efficiencies and about one order of magnitude higher than the efficiency for equivalent SLM-generated beams. The good quality of the generated versus the SLM-generated beams is confirmed by a root mean squared deviation of 0.08 and 0.06 in the normalized intensity modulation of the Bessel and Weber beams, respectively, which were probed to be non-diffracting over an extended region of several centimeters. The transparency of the device in the visible range and the well known achromaticity of the polarization holograms [1

1. L. Nikolova and P. S. Ramanujam, Polarization Holography, (Cambridge University Press, 2009).

] make these elements an excellent alternative to generate complex beams, such as propagation invariant optical fields. The polarization properties of the diffracted beams make them suitable to obtain some kinds of vector beams by direct recombination [22

22. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007). [CrossRef]

, 23

23. A. Flores-Pérez, J. Hernández-Hernández, R. Jáuregui, and K. Volke-Sepúlveda, “Experimental generation and analysis of first-order TE and TM Bessel modes in free space,” Opt. Lett. 31(11), 1732–1734 (2006). [CrossRef] [PubMed]

]. Moreover, the PHs we have presented here, are the only optical elements that can store polarization, phase and amplitude information simultaneously. This characteristic makes them attractive for applications in quantum information.

Acknowledgments

This work was supported by the Cooperation Project of Great Relevance Italy-Mexico 2012 of the Italian Foreign Ministry. It was co-funded by the European Commission, SFE, and Regione Calabria. The authors wish to thank V. P. Shibaev for the synthesis of the amorphous azopolymer and for fruitful discussions.

References and links

1.

L. Nikolova and P. S. Ramanujam, Polarization Holography, (Cambridge University Press, 2009).

2.

C. Provenzano, P. Pagliusi, and G. Cipparrone, “Highly efficient liquid crystal based diffraction grating induced by polarization holograms at the aligning surfaces,” Appl. Phys. Lett. 89(12), 121105 (2006). [CrossRef]

3.

E. Nicolescu and M. J. Escuti, “Polarization-independent tunable optical filters using bilayer polarization gratings,” Appl. Opt. 49(20), 3900–3904 (2010). [CrossRef] [PubMed]

4.

S. R. Nersisyan, N. V. Tabiryan, D. M. Steeves, and B. R. Kimball, “Optical axis gratings in liquid crystals and their use for polarization insensitive optical switching,” J. Nonlinear Opt. Phys. Mater. 18(01), 1–47 (2009). [CrossRef]

5.

H. Choi, J. H. Woo, J. W. Wu, D. W. Kim, T. K. Lim, and S. H. Song, “Holographic inscription of helical wavefronts in a liquid crystal polarization grating,” Appl. Phys. Lett. 91(14), 141112 (2007). [CrossRef]

6.

Y. Li, J. Kim, and M. J. Escuti, “Controlling orbital angular momentum using forked polarization gratings,” Proc. SPIE 7789, 77890F77890F-12 (2010). [CrossRef]

7.

M. Fratz, P. Fischer, and D. M. Giel, “Full phase and amplitude control in computer-generated holography,” Opt. Lett. 34(23), 3659–3661 (2009). [CrossRef] [PubMed]

8.

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). [CrossRef] [PubMed]

9.

D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005). [CrossRef]

10.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000). [CrossRef] [PubMed]

11.

M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004). [CrossRef] [PubMed]

12.

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). [CrossRef] [PubMed]

13.

K. Volke-Sepúlveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclassical Opt. 4(2), S82–S89 (2002). [CrossRef]

14.

V. Garcés-Chávez, K. Volke-Sepúlveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A 66(6), 063402 (2002). [CrossRef]

15.

J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]

16.

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, “Shaping soliton properties in Mathieu lattices,” Opt. Lett. 31(2), 238–240 (2006). [CrossRef] [PubMed]

17.

A. Ruelas, S. Lopez-Aguayo, and J. C. Gutiérrez-Vega, “Stable solitons in elliptical photonic lattices,” Opt. Lett. 33(23), 2785–2787 (2008). [CrossRef] [PubMed]

18.

B. M. Rodriguez-Lara and R. Jauregui, “Dynamical constants for electromagnetic fields with elliptic-cylindrical symmetry,” Phys. Rev. A 78(3), 033813 (2008). [CrossRef]

19.

B. M. Rodriguez-Lara and R. Jauregui, “A single structured light beam as an atomic cloud splitter,” Phys. Rev. A 80(1), 011813 (2009). [CrossRef]

20.

V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated Phase Computer Holograms for the Accurate Encoding of Scalar Complex Fields,” J. Opt. Soc. Am. A 24(11), 3500–3507 (2007). [CrossRef] [PubMed]

21.

G. Cipparrone, P. Pagliusi, C. Provenzano, and V. P. Shibaev, “Polarization holographic recording in amorphous polymer with photoinduced linear and circular birefringence,” J. Phys. Chem. B 114(27), 8900–8904 (2010). [CrossRef] [PubMed]

22.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007). [CrossRef]

23.

A. Flores-Pérez, J. Hernández-Hernández, R. Jáuregui, and K. Volke-Sepúlveda, “Experimental generation and analysis of first-order TE and TM Bessel modes in free space,” Opt. Lett. 31(11), 1732–1734 (2006). [CrossRef] [PubMed]

OCIS Codes
(050.1970) Diffraction and gratings : Diffractive optics
(090.2890) Holography : Holographic optical elements
(090.2900) Holography : Optical storage materials
(230.5440) Optical devices : Polarization-selective devices
(230.6120) Optical devices : Spatial light modulators
(070.3185) Fourier optics and signal processing : Invariant optical fields
(050.4865) Diffraction and gratings : Optical vortices

ToC Category:
Holography

History
Original Manuscript: November 12, 2012
Revised Manuscript: December 10, 2012
Manuscript Accepted: December 11, 2012
Published: March 19, 2013

Citation
U. Ruiz, P. Pagliusi, C. Provenzano, K. Volke-Sepúlveda, and Gabriella Cipparrone, "Polarization holograms allow highly efficient generation of complex light beams," Opt. Express 21, 7505-7510 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-6-7505


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References

  1. L. Nikolova and P. S. Ramanujam, Polarization Holography, (Cambridge University Press, 2009).
  2. C. Provenzano, P. Pagliusi, and G. Cipparrone, “Highly efficient liquid crystal based diffraction grating induced by polarization holograms at the aligning surfaces,” Appl. Phys. Lett.89(12), 121105 (2006). [CrossRef]
  3. E. Nicolescu and M. J. Escuti, “Polarization-independent tunable optical filters using bilayer polarization gratings,” Appl. Opt.49(20), 3900–3904 (2010). [CrossRef] [PubMed]
  4. S. R. Nersisyan, N. V. Tabiryan, D. M. Steeves, and B. R. Kimball, “Optical axis gratings in liquid crystals and their use for polarization insensitive optical switching,” J. Nonlinear Opt. Phys. Mater.18(01), 1–47 (2009). [CrossRef]
  5. H. Choi, J. H. Woo, J. W. Wu, D. W. Kim, T. K. Lim, and S. H. Song, “Holographic inscription of helical wavefronts in a liquid crystal polarization grating,” Appl. Phys. Lett.91(14), 141112 (2007). [CrossRef]
  6. Y. Li, J. Kim, and M. J. Escuti, “Controlling orbital angular momentum using forked polarization gratings,” Proc. SPIE7789, 77890F77890F-12 (2010). [CrossRef]
  7. M. Fratz, P. Fischer, and D. M. Giel, “Full phase and amplitude control in computer-generated holography,” Opt. Lett.34(23), 3659–3661 (2009). [CrossRef] [PubMed]
  8. 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). [CrossRef] [PubMed]
  9. D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys.46(1), 15–28 (2005). [CrossRef]
  10. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett.25(20), 1493–1495 (2000). [CrossRef] [PubMed]
  11. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett.29(1), 44–46 (2004). [CrossRef] [PubMed]
  12. 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. A45(11), 8185–8189 (1992). [CrossRef] [PubMed]
  13. K. Volke-Sepúlveda, V. Garcés-Chávez, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B Quantum Semiclassical Opt.4(2), S82–S89 (2002). [CrossRef]
  14. V. Garcés-Chávez, K. Volke-Sepúlveda, S. Chávez-Cerda, W. Sibbett, and K. Dholakia, “Transfer of orbital angular momentum to an optically trapped low-index particle,” Phys. Rev. A66(6), 063402 (2002). [CrossRef]
  15. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987). [CrossRef] [PubMed]
  16. Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, “Shaping soliton properties in Mathieu lattices,” Opt. Lett.31(2), 238–240 (2006). [CrossRef] [PubMed]
  17. A. Ruelas, S. Lopez-Aguayo, and J. C. Gutiérrez-Vega, “Stable solitons in elliptical photonic lattices,” Opt. Lett.33(23), 2785–2787 (2008). [CrossRef] [PubMed]
  18. B. M. Rodriguez-Lara and R. Jauregui, “Dynamical constants for electromagnetic fields with elliptic-cylindrical symmetry,” Phys. Rev. A78(3), 033813 (2008). [CrossRef]
  19. B. M. Rodriguez-Lara and R. Jauregui, “A single structured light beam as an atomic cloud splitter,” Phys. Rev. A80(1), 011813 (2009). [CrossRef]
  20. V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated Phase Computer Holograms for the Accurate Encoding of Scalar Complex Fields,” J. Opt. Soc. Am. A24(11), 3500–3507 (2007). [CrossRef] [PubMed]
  21. G. Cipparrone, P. Pagliusi, C. Provenzano, and V. P. Shibaev, “Polarization holographic recording in amorphous polymer with photoinduced linear and circular birefringence,” J. Phys. Chem. B114(27), 8900–8904 (2010). [CrossRef] [PubMed]
  22. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys.9(3), 78 (2007). [CrossRef]
  23. A. Flores-Pérez, J. Hernández-Hernández, R. Jáuregui, and K. Volke-Sepúlveda, “Experimental generation and analysis of first-order TE and TM Bessel modes in free space,” Opt. Lett.31(11), 1732–1734 (2006). [CrossRef] [PubMed]

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