Abstract

This work demonstrates experimental approaches to characterize a single multimode fiber imaging system without a reference beam. Spatial light modulation is performed with a digital micro-mirror device that enables high-speed binary amplitude modulation. Intensity-only images are recorded by the camera and processed by a Bayesian inference based algorithm to retrieve the phase of the output optical field as well as the transmission matrix of the fiber. The calculated transmission matrix is validated by three standards: prediction accuracy, transmission imaging, and focus generation. Also, it is found that information on mode count and eigenchannels can be extracted from the transmission matrix by singular value decomposition. This paves the way for a more compact and cheaper single multimode fiber imaging system for many demanding imaging tasks.

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

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References

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  1. D. Psaltis and C. Moser, “Imaging with multimode fibers,” Opt. Photonics News 27(1), 24–31 (2016).
    [Crossref]
  2. S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
    [Crossref] [PubMed]
  3. S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1(6), 81 (2010).
    [Crossref] [PubMed]
  4. Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
    [Crossref] [PubMed]
  5. I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “Focusing and scanning light through a multimode optical fiber using digital phase conjugation,” Opt. Express 20(10), 10583–10590 (2012).
    [Crossref] [PubMed]
  6. I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “High-resolution, lensless endoscope based on digital scanning through a multimode optical fiber,” Biomed. Opt. Express 4(2), 260–270 (2013).
    [Crossref] [PubMed]
  7. D. Loterie, S. Farahi, I. Papadopoulos, A. Goy, D. Psaltis, and C. Moser, “Digital confocal microscopy through a multimode fiber,” Opt. Express 23(18), 23845–23858 (2015).
    [Crossref] [PubMed]
  8. D. Loterie, S. A. Goorden, D. Psaltis, and C. Moser, “Confocal microscopy through a multimode fiber using optical correlation,” Opt. Lett. 40(24), 5754–5757 (2015).
    [Crossref] [PubMed]
  9. D. Kim, J. Moon, M. Kim, T. D. Yang, J. Kim, E. Chung, and W. Choi, “Toward a miniature endomicroscope: pixelation-free and diffraction-limited imaging through a fiber bundle,” Opt. Lett. 39(7), 1921–1924 (2014).
    [Crossref] [PubMed]
  10. A. M. Caravaca-Aguirre and R. Piestun, “Single multimode fiber endoscope,” Opt. Express 25(3), 1656–1665 (2017).
    [Crossref] [PubMed]
  11. D. Loterie, D. Psaltis, and C. Moser, “Bend translation in multimode fiber imaging,” Opt. Express 25(6), 6263–6273 (2017).
    [Crossref] [PubMed]
  12. S. Farahi, D. Ziegler, I. N. Papadopoulos, D. Psaltis, and C. Moser, “Dynamic bending compensation while focusing through a multimode fiber,” Opt. Express 21(19), 22504–22514 (2013).
    [Crossref] [PubMed]
  13. M. Plöschner, T. Tyc, and T. Cizmar, “Seeing through chaos in multimode fibres,” Nat. Photonics 9(8), 529–535 (2015).
    [Crossref]
  14. A. M. Caravaca-Aguirre, E. Niv, D. B. Conkey, and R. Piestun, “Real-time resilient focusing through a bending multimode fiber,” Opt. Express 21(10), 12881–12887 (2013).
    [Crossref] [PubMed]
  15. A. Drémeau, A. Liutkus, D. Martina, O. Katz, C. Schülke, F. Krzakala, S. Gigan, and L. Daudet, “Reference-less measurement of the transmission matrix of a highly scattering material using a DMD and phase retrieval techniques,” Opt. Express 23(9), 11898–11911 (2015).
    [Crossref] [PubMed]
  16. D. Kogan, S. Sivankutty, V. Tsvirkun, G. Bouwmans, E. R. Andresen, H. Rigneault, and D. Oron, “Phase retrieval in multicore fiber bundles,” Opt. Lett. 42(3), 647–650 (2017).
    [Crossref] [PubMed]
  17. A. Fertman and D. Yelin, “Image transmission through an optical fiber using real-time modal phase restoration,” J. Opt. Soc. Am. B 30(1), 149–157 (2013).
    [Crossref]
  18. M. Gom, T. B. Norris, E. Michielssen, and R. R. Nadakuditi, “Mode control in a multimode fiber through acquiring its transmission matrix from a reference-less optical system,” Opt. Lett. 43(3), 419–422 (2018).
    [Crossref] [PubMed]
  19. C. Yoon, Y. Choi, M. Kim, J. Moon, D. Kim, and W. Choi, “Experimental measurement of the number of modes for a multimode optical fiber,” Opt. Lett. 37(21), 4558–4560 (2012).
    [Crossref] [PubMed]
  20. M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6(9), 581–585 (2012).
    [Crossref]

2018 (1)

2017 (3)

2016 (1)

D. Psaltis and C. Moser, “Imaging with multimode fibers,” Opt. Photonics News 27(1), 24–31 (2016).
[Crossref]

2015 (4)

2014 (1)

2013 (4)

2012 (4)

C. Yoon, Y. Choi, M. Kim, J. Moon, D. Kim, and W. Choi, “Experimental measurement of the number of modes for a multimode optical fiber,” Opt. Lett. 37(21), 4558–4560 (2012).
[Crossref] [PubMed]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6(9), 581–585 (2012).
[Crossref]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[Crossref] [PubMed]

I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “Focusing and scanning light through a multimode optical fiber using digital phase conjugation,” Opt. Express 20(10), 10583–10590 (2012).
[Crossref] [PubMed]

2010 (2)

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[Crossref] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1(6), 81 (2010).
[Crossref] [PubMed]

Andresen, E. R.

Boccara, A. C.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[Crossref] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1(6), 81 (2010).
[Crossref] [PubMed]

Bouwmans, G.

Caravaca-Aguirre, A. M.

Carminati, R.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[Crossref] [PubMed]

Choi, W.

D. Kim, J. Moon, M. Kim, T. D. Yang, J. Kim, E. Chung, and W. Choi, “Toward a miniature endomicroscope: pixelation-free and diffraction-limited imaging through a fiber bundle,” Opt. Lett. 39(7), 1921–1924 (2014).
[Crossref] [PubMed]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[Crossref] [PubMed]

C. Yoon, Y. Choi, M. Kim, J. Moon, D. Kim, and W. Choi, “Experimental measurement of the number of modes for a multimode optical fiber,” Opt. Lett. 37(21), 4558–4560 (2012).
[Crossref] [PubMed]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6(9), 581–585 (2012).
[Crossref]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6(9), 581–585 (2012).
[Crossref]

Choi, Y.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6(9), 581–585 (2012).
[Crossref]

C. Yoon, Y. Choi, M. Kim, J. Moon, D. Kim, and W. Choi, “Experimental measurement of the number of modes for a multimode optical fiber,” Opt. Lett. 37(21), 4558–4560 (2012).
[Crossref] [PubMed]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[Crossref] [PubMed]

Chung, E.

Cizmar, T.

M. Plöschner, T. Tyc, and T. Cizmar, “Seeing through chaos in multimode fibres,” Nat. Photonics 9(8), 529–535 (2015).
[Crossref]

Conkey, D. B.

Dasari, R. R.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[Crossref] [PubMed]

Daudet, L.

Drémeau, A.

Fang-Yen, C.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[Crossref] [PubMed]

Farahi, S.

Fertman, A.

Fink, M.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1(6), 81 (2010).
[Crossref] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[Crossref] [PubMed]

Gigan, S.

A. Drémeau, A. Liutkus, D. Martina, O. Katz, C. Schülke, F. Krzakala, S. Gigan, and L. Daudet, “Reference-less measurement of the transmission matrix of a highly scattering material using a DMD and phase retrieval techniques,” Opt. Express 23(9), 11898–11911 (2015).
[Crossref] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[Crossref] [PubMed]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1(6), 81 (2010).
[Crossref] [PubMed]

Gom, M.

Goorden, S. A.

Goy, A.

Katz, O.

Kim, D.

Kim, J.

D. Kim, J. Moon, M. Kim, T. D. Yang, J. Kim, E. Chung, and W. Choi, “Toward a miniature endomicroscope: pixelation-free and diffraction-limited imaging through a fiber bundle,” Opt. Lett. 39(7), 1921–1924 (2014).
[Crossref] [PubMed]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6(9), 581–585 (2012).
[Crossref]

Kim, M.

D. Kim, J. Moon, M. Kim, T. D. Yang, J. Kim, E. Chung, and W. Choi, “Toward a miniature endomicroscope: pixelation-free and diffraction-limited imaging through a fiber bundle,” Opt. Lett. 39(7), 1921–1924 (2014).
[Crossref] [PubMed]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[Crossref] [PubMed]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6(9), 581–585 (2012).
[Crossref]

C. Yoon, Y. Choi, M. Kim, J. Moon, D. Kim, and W. Choi, “Experimental measurement of the number of modes for a multimode optical fiber,” Opt. Lett. 37(21), 4558–4560 (2012).
[Crossref] [PubMed]

Kogan, D.

Krzakala, F.

Lee, K. J.

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[Crossref] [PubMed]

Lerosey, G.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1(6), 81 (2010).
[Crossref] [PubMed]

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[Crossref] [PubMed]

Liutkus, A.

Loterie, D.

Martina, D.

Michielssen, E.

Moon, J.

Moser, C.

Nadakuditi, R. R.

Niv, E.

Norris, T. B.

Oron, D.

Papadopoulos, I.

Papadopoulos, I. N.

Park, Q.-H.

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6(9), 581–585 (2012).
[Crossref]

Piestun, R.

Plöschner, M.

M. Plöschner, T. Tyc, and T. Cizmar, “Seeing through chaos in multimode fibres,” Nat. Photonics 9(8), 529–535 (2015).
[Crossref]

Popoff, S.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1(6), 81 (2010).
[Crossref] [PubMed]

Popoff, S. M.

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[Crossref] [PubMed]

Psaltis, D.

Rigneault, H.

Schülke, C.

Sivankutty, S.

Tsvirkun, V.

Tyc, T.

M. Plöschner, T. Tyc, and T. Cizmar, “Seeing through chaos in multimode fibres,” Nat. Photonics 9(8), 529–535 (2015).
[Crossref]

Yang, T. D.

D. Kim, J. Moon, M. Kim, T. D. Yang, J. Kim, E. Chung, and W. Choi, “Toward a miniature endomicroscope: pixelation-free and diffraction-limited imaging through a fiber bundle,” Opt. Lett. 39(7), 1921–1924 (2014).
[Crossref] [PubMed]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[Crossref] [PubMed]

Yelin, D.

Yoon, C.

C. Yoon, Y. Choi, M. Kim, J. Moon, D. Kim, and W. Choi, “Experimental measurement of the number of modes for a multimode optical fiber,” Opt. Lett. 37(21), 4558–4560 (2012).
[Crossref] [PubMed]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6(9), 581–585 (2012).
[Crossref]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[Crossref] [PubMed]

Ziegler, D.

Biomed. Opt. Express (1)

J. Opt. Soc. Am. B (1)

Nat. Commun. (1)

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1(6), 81 (2010).
[Crossref] [PubMed]

Nat. Photonics (2)

M. Plöschner, T. Tyc, and T. Cizmar, “Seeing through chaos in multimode fibres,” Nat. Photonics 9(8), 529–535 (2015).
[Crossref]

M. Kim, Y. Choi, C. Yoon, W. Choi, J. Kim, Q.-H. Park, and W. Choi, “Maximal energy transport through disordered media with the implementation of transmission eigenchannels,” Nat. Photonics 6(9), 581–585 (2012).
[Crossref]

Opt. Express (7)

Opt. Lett. (5)

Opt. Photonics News (1)

D. Psaltis and C. Moser, “Imaging with multimode fibers,” Opt. Photonics News 27(1), 24–31 (2016).
[Crossref]

Phys. Rev. Lett. (2)

S. M. Popoff, G. Lerosey, R. Carminati, M. Fink, A. C. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: An approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104(10), 100601 (2010).
[Crossref] [PubMed]

Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett. 109(20), 203901 (2012).
[Crossref] [PubMed]

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Figures (8)

Fig. 1
Fig. 1 Experimental setup of reference-less MMF characterization. M1, M2: Mirrors; L1, L2: Bi-Convex Lenses; DMD: Digital Micromirror Device; TL1, TL2: Tube Lens; OL1, OL2: Objective Lens; MMF: Multimode Fiber; P: Polarizer; OP: Object Plane; IP: Image Plane.
Fig. 2
Fig. 2 Prediction performance expressed by the correlation coefficient as a function of P/N. N is the count of macro pixels used on DMD while P denotes the number of measurements performed during calibration.
Fig. 3
Fig. 3 MMF Imaging in transmission mode. (a-c) Original patterns; (d-f) Reconstructed images. Red boxes indicate the regions used for resolution evaluation. Scale Bar in (f): 5 μm; (g-i) Amplitude line profiles for the dashed blue lines in the images above.
Fig. 4
Fig. 4 Focus generation with the TM information. (a) On axis focus. Scale bar: 10 μm; (b) Normalized EF distribution across a 50 by 50 grid within the distal fiber end. Grid pitch: 0.58 μm; (c, d) Intensity profiles of the focus in (a) along the y and x axes respectively.
Fig. 5
Fig. 5 Comparison of distribution of normalized squared singular values of TM and normalized transmittance (a) Variation of normalized squared singular values with fiber mode index; (b) Variation of normalized transmittance with eigenchannel index.
Fig. 6
Fig. 6 Observation of fiber mode count change. (a) Change of fiber mode count over adjustment of the translation stage. (b) Fiber bending experiment configuration. MMF: Multimode Fiber, FS: Fiber Stage, TS: Translation Stage, R: Radius of the circle formed by fiber (around 20 cm). Yellow arrow indicates the translation direction of TS.
Fig. 7
Fig. 7 Normalized transmittance for random input patterns.
Fig. 8
Fig. 8 Comparison of the ideal and experimental eigenchannels. (a) Ideal input amplitude; (b) DMD reflectivity input amplitude; (c) Theoretical output pattern; (d) Experimental output pattern.

Equations (21)

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

y=| Dx |.
y u = e j θ u ( i=1 N d ui x i * + n u ) ,
p(x)= i=1 N p( x i ) ,
p( x i )=CN(0, σ x 2 ),
p(θ)= u=1 M p( θ u ),
p( θ u )= 1 2π ,
x ^ = argmax x p(x|y),
p(x|y)= θ p(x,θ|y) .
q ^ (x,θ)= i=1 N q i ( x i ) u=1 M q u ( θ u ) ,
q( x i )=CN( m i , i ),
i = σ n 2 σ x 2 σ n 2 + σ x 2 d i H d i ,
m i = σ x 2 σ n 2 + σ x 2 d i H d i r i H d i ,
r i = y ¯ ki m k * d k ,
y ¯ =[ y u e jarg( y u * z u ) I 1 ( 2 σ n 2 | y u * z u |) I 0 ( 2 σ n 2 | y u * z u |) ],
z u = i=1 N m i d ui ,
p(x|y)= θ p(x,θ|y) θ q(x,θ) θ i q( x i ) u q( θ u ) i q( x i ) ,
x ^ i = argmax x i q( x i )= m i ,
TM E OP = E IP ,
N m 0.5× F 2 ,
F=2πaNA/λ,
TM=US V * .

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