Abstract

A simple and flexible photonic approach to generating a triangular microwave waveform using a single integrated polarization-multiplexing dual-drive Mach-Zehnder modulator (PM-DMZM) and a polarizer is proposed and demonstrated, which needs no specific large modulation indices or an optical filter. In the proposed method, one sub-Mach-Zehnder modulator (MZM) in the PM-DMZM is driven by a fundamental frequency, which generates an optical signal composed of an optical carrier and a + 1st-order sideband along one polarization direction; and the other sub-MZM is driven by a frequency tripled signal, generating an optical carrier and a −1st-order sideband along the orthogonal polarization direction. By adjusting the polarization direction of the polarizer following the PM-DMZM, which changes the power ratio of the two sidebands, optical intensity with expression corresponding to the Fourier expansion of a triangular-shaped waveform is obtained. Different from the previously reported approaches, neither specific large modulation index nor optical filtering is required, which guarantees a large operational frequency range and improved robustness. A proof-of-concept experiment is carried out. 5-GHz triangular-shaped waveform signals are successfully generated with different modulation indices.

© 2016 Optical Society of America

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References

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  1. M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
    [Crossref]
  2. A. I. Latkin, S. Boscolo, R. S. Bhamber, and S. K. Turitsyn, “Optical frequency conversion, pulse compression and signal copying using triangular pulses,” in ECOC 2008 (2008), paper Th1B.2.
  3. J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photonics Technol. Lett. 15(4), 93–96 (2002).
  4. J. Ye, L. Yan, W. Pan, B. Luo, X. Zou, A. Yi, and S. Yao, “Photonic generation of triangular-shaped pulses based on frequency-to-time conversion,” Opt. Lett. 36(8), 1458–1460 (2011).
    [Crossref] [PubMed]
  5. H. Y. Jiang, L. S. Yan, Y. F. Sun, J. Ye, W. Pan, B. Luo, and X. H. Zou, “Photonic arbitrary waveform generation based on crossed frequency to time mapping,” Opt. Express 21(5), 6488–6496 (2013).
    [Crossref] [PubMed]
  6. B. Dai, Z. Gao, X. Wang, H. Chen, N. Kataoka, and N. Wada, “Generation of versatile waveforms from CW light using a dual-drive Mach-Zehnder modulator and employing chromatic dispersion,” J. Lightwave Technol. 31(1), 145–151 (2013).
    [Crossref]
  7. Z. Wu, L. Lei, J. Dong, and X. Zhang, “Triangular-shaped pulse generation based on self-convolution of a rectangular-shaped pulse,” Opt. Lett. 39(8), 2258–2261 (2014).
    [Crossref] [PubMed]
  8. W. Li, W. T. Wang, W. H. Sun, W. Y. Wang, and N. H. Zhu, “Generation of triangular waveforms based on a microwave photonic filter with negative coefficient,” Opt. Express 22(12), 14993–15001 (2014).
    [Crossref] [PubMed]
  9. J. Li, J. Sun, W. Xu, T. Ning, L. Pei, J. Yuan, and Y. Li, “Frequency-doubled triangular-shaped waveform generation based on spectrum manipulation,” Opt. Lett. 41(2), 199–202 (2016).
    [Crossref] [PubMed]
  10. W. Li, W. T. Wang, and N. H. Zhu, “Photonic generation of radio-frequency waveforms based on dual-parallel Mach–Zehnder modulator,” IEEE Photonics J. 6(3), 1–8 (2014).
    [Crossref]
  11. X. Liu, W. Pan, X. Zou, D. Zheng, L. Yan, B. Luo, and B. Lu, “Photonic generation of triangular-shaped microwave pulses using SBS-based optical carrier processing,” J. Lightwave Technol. 32(20), 3797–3802 (2014).
    [Crossref]
  12. W. Liu and J. Yao, “Photonic generation of microwave waveforms based on a polarization modulator in a Sagnac Loop,” J. Lightwave Technol. 32(20), 3637–3644 (2014).
    [Crossref]
  13. J. Li, X. Zhang, B. Hraimel, T. Ning, L. Pei, and K. Wu, “Performance analysis of a photonic-assisted periodic triangular-shaped pulses generator,” J. Lightwave Technol. 30(11), 1617–1624 (2012).
    [Crossref]
  14. F. Zhang, X. Ge, and S. Pan, “Triangular pulse generation using a dual-parallel Mach-Zehnder modulator driven by a single-frequency radio frequency signal,” Opt. Lett. 38(21), 4491–4493 (2013).
    [Crossref] [PubMed]
  15. J. Li, T. Ning, L. Pei, W. Peng, N. Jia, Q. Zhou, and X. Wen, “Photonic generation of triangular waveform signals by using a dual-parallel Mach-Zehnder modulator,” Opt. Lett. 36(19), 3828–3830 (2011).
    [Crossref] [PubMed]
  16. F. Zhang, B. Gao, P. Zhou, and S. Pan, “Triangular pulse generation by polarization multiplexed optoelectronic oscillator,” IEEE Photonics Technol. Lett. 28(15), 1645–1648 (2016).
    [Crossref]
  17. L. Huang, D. Chen, P. Wang, T. Zhang, P. Xiang, Y. Zhang, T. Pu, and X. Chen, “Generation of triangular pulses based on an optoelectronic oscillator,” IEEE Photonics Technol. Lett. 27(23), 2500–2503 (2015).
    [Crossref]
  18. Z. Li, H. Chi, X. Zhang, and J. Yao, “Optical single-sideband modulation using a fiber-Bragg-grating-based optical Hilbert transformer,” IEEE Photonics Technol. Lett. 23(9), 558–560 (2011).
    [Crossref]

2016 (2)

F. Zhang, B. Gao, P. Zhou, and S. Pan, “Triangular pulse generation by polarization multiplexed optoelectronic oscillator,” IEEE Photonics Technol. Lett. 28(15), 1645–1648 (2016).
[Crossref]

J. Li, J. Sun, W. Xu, T. Ning, L. Pei, J. Yuan, and Y. Li, “Frequency-doubled triangular-shaped waveform generation based on spectrum manipulation,” Opt. Lett. 41(2), 199–202 (2016).
[Crossref] [PubMed]

2015 (1)

L. Huang, D. Chen, P. Wang, T. Zhang, P. Xiang, Y. Zhang, T. Pu, and X. Chen, “Generation of triangular pulses based on an optoelectronic oscillator,” IEEE Photonics Technol. Lett. 27(23), 2500–2503 (2015).
[Crossref]

2014 (5)

2013 (3)

2012 (1)

2011 (3)

2010 (1)

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

2002 (1)

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photonics Technol. Lett. 15(4), 93–96 (2002).

Chen, D.

L. Huang, D. Chen, P. Wang, T. Zhang, P. Xiang, Y. Zhang, T. Pu, and X. Chen, “Generation of triangular pulses based on an optoelectronic oscillator,” IEEE Photonics Technol. Lett. 27(23), 2500–2503 (2015).
[Crossref]

Chen, H.

Chen, X.

L. Huang, D. Chen, P. Wang, T. Zhang, P. Xiang, Y. Zhang, T. Pu, and X. Chen, “Generation of triangular pulses based on an optoelectronic oscillator,” IEEE Photonics Technol. Lett. 27(23), 2500–2503 (2015).
[Crossref]

Chi, H.

Z. Li, H. Chi, X. Zhang, and J. Yao, “Optical single-sideband modulation using a fiber-Bragg-grating-based optical Hilbert transformer,” IEEE Photonics Technol. Lett. 23(9), 558–560 (2011).
[Crossref]

Chou, J.

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photonics Technol. Lett. 15(4), 93–96 (2002).

Dai, B.

Dong, J.

Gao, B.

F. Zhang, B. Gao, P. Zhou, and S. Pan, “Triangular pulse generation by polarization multiplexed optoelectronic oscillator,” IEEE Photonics Technol. Lett. 28(15), 1645–1648 (2016).
[Crossref]

Gao, Z.

Ge, X.

Han, Y.

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photonics Technol. Lett. 15(4), 93–96 (2002).

Hraimel, B.

Huang, L.

L. Huang, D. Chen, P. Wang, T. Zhang, P. Xiang, Y. Zhang, T. Pu, and X. Chen, “Generation of triangular pulses based on an optoelectronic oscillator,” IEEE Photonics Technol. Lett. 27(23), 2500–2503 (2015).
[Crossref]

Jalali, B.

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photonics Technol. Lett. 15(4), 93–96 (2002).

Jia, N.

Jiang, H. Y.

Kataoka, N.

Khan, M. H.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Leaird, D. E.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Lei, L.

Li, J.

Li, W.

W. Li, W. T. Wang, and N. H. Zhu, “Photonic generation of radio-frequency waveforms based on dual-parallel Mach–Zehnder modulator,” IEEE Photonics J. 6(3), 1–8 (2014).
[Crossref]

W. Li, W. T. Wang, W. H. Sun, W. Y. Wang, and N. H. Zhu, “Generation of triangular waveforms based on a microwave photonic filter with negative coefficient,” Opt. Express 22(12), 14993–15001 (2014).
[Crossref] [PubMed]

Li, Y.

Li, Z.

Z. Li, H. Chi, X. Zhang, and J. Yao, “Optical single-sideband modulation using a fiber-Bragg-grating-based optical Hilbert transformer,” IEEE Photonics Technol. Lett. 23(9), 558–560 (2011).
[Crossref]

Liu, W.

Liu, X.

Lu, B.

Luo, B.

Ning, T.

Pan, S.

F. Zhang, B. Gao, P. Zhou, and S. Pan, “Triangular pulse generation by polarization multiplexed optoelectronic oscillator,” IEEE Photonics Technol. Lett. 28(15), 1645–1648 (2016).
[Crossref]

F. Zhang, X. Ge, and S. Pan, “Triangular pulse generation using a dual-parallel Mach-Zehnder modulator driven by a single-frequency radio frequency signal,” Opt. Lett. 38(21), 4491–4493 (2013).
[Crossref] [PubMed]

Pan, W.

Pei, L.

Peng, W.

Pu, T.

L. Huang, D. Chen, P. Wang, T. Zhang, P. Xiang, Y. Zhang, T. Pu, and X. Chen, “Generation of triangular pulses based on an optoelectronic oscillator,” IEEE Photonics Technol. Lett. 27(23), 2500–2503 (2015).
[Crossref]

Qi, M.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Shen, H.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Sun, J.

Sun, W. H.

Sun, Y. F.

Wada, N.

Wang, P.

L. Huang, D. Chen, P. Wang, T. Zhang, P. Xiang, Y. Zhang, T. Pu, and X. Chen, “Generation of triangular pulses based on an optoelectronic oscillator,” IEEE Photonics Technol. Lett. 27(23), 2500–2503 (2015).
[Crossref]

Wang, W. T.

W. Li, W. T. Wang, and N. H. Zhu, “Photonic generation of radio-frequency waveforms based on dual-parallel Mach–Zehnder modulator,” IEEE Photonics J. 6(3), 1–8 (2014).
[Crossref]

W. Li, W. T. Wang, W. H. Sun, W. Y. Wang, and N. H. Zhu, “Generation of triangular waveforms based on a microwave photonic filter with negative coefficient,” Opt. Express 22(12), 14993–15001 (2014).
[Crossref] [PubMed]

Wang, W. Y.

Wang, X.

Weiner, A. M.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Wen, X.

Wu, K.

Wu, Z.

Xiang, P.

L. Huang, D. Chen, P. Wang, T. Zhang, P. Xiang, Y. Zhang, T. Pu, and X. Chen, “Generation of triangular pulses based on an optoelectronic oscillator,” IEEE Photonics Technol. Lett. 27(23), 2500–2503 (2015).
[Crossref]

Xiao, S.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Xu, W.

Xuan, Y.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Yan, L.

Yan, L. S.

Yao, J.

W. Liu and J. Yao, “Photonic generation of microwave waveforms based on a polarization modulator in a Sagnac Loop,” J. Lightwave Technol. 32(20), 3637–3644 (2014).
[Crossref]

Z. Li, H. Chi, X. Zhang, and J. Yao, “Optical single-sideband modulation using a fiber-Bragg-grating-based optical Hilbert transformer,” IEEE Photonics Technol. Lett. 23(9), 558–560 (2011).
[Crossref]

Yao, S.

Ye, J.

Yi, A.

Yuan, J.

Zhang, F.

F. Zhang, B. Gao, P. Zhou, and S. Pan, “Triangular pulse generation by polarization multiplexed optoelectronic oscillator,” IEEE Photonics Technol. Lett. 28(15), 1645–1648 (2016).
[Crossref]

F. Zhang, X. Ge, and S. Pan, “Triangular pulse generation using a dual-parallel Mach-Zehnder modulator driven by a single-frequency radio frequency signal,” Opt. Lett. 38(21), 4491–4493 (2013).
[Crossref] [PubMed]

Zhang, T.

L. Huang, D. Chen, P. Wang, T. Zhang, P. Xiang, Y. Zhang, T. Pu, and X. Chen, “Generation of triangular pulses based on an optoelectronic oscillator,” IEEE Photonics Technol. Lett. 27(23), 2500–2503 (2015).
[Crossref]

Zhang, X.

Zhang, Y.

L. Huang, D. Chen, P. Wang, T. Zhang, P. Xiang, Y. Zhang, T. Pu, and X. Chen, “Generation of triangular pulses based on an optoelectronic oscillator,” IEEE Photonics Technol. Lett. 27(23), 2500–2503 (2015).
[Crossref]

Zhao, L.

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Zheng, D.

Zhou, P.

F. Zhang, B. Gao, P. Zhou, and S. Pan, “Triangular pulse generation by polarization multiplexed optoelectronic oscillator,” IEEE Photonics Technol. Lett. 28(15), 1645–1648 (2016).
[Crossref]

Zhou, Q.

Zhu, N. H.

W. Li, W. T. Wang, and N. H. Zhu, “Photonic generation of radio-frequency waveforms based on dual-parallel Mach–Zehnder modulator,” IEEE Photonics J. 6(3), 1–8 (2014).
[Crossref]

W. Li, W. T. Wang, W. H. Sun, W. Y. Wang, and N. H. Zhu, “Generation of triangular waveforms based on a microwave photonic filter with negative coefficient,” Opt. Express 22(12), 14993–15001 (2014).
[Crossref] [PubMed]

Zou, X.

Zou, X. H.

IEEE Photonics J. (1)

W. Li, W. T. Wang, and N. H. Zhu, “Photonic generation of radio-frequency waveforms based on dual-parallel Mach–Zehnder modulator,” IEEE Photonics J. 6(3), 1–8 (2014).
[Crossref]

IEEE Photonics Technol. Lett. (4)

F. Zhang, B. Gao, P. Zhou, and S. Pan, “Triangular pulse generation by polarization multiplexed optoelectronic oscillator,” IEEE Photonics Technol. Lett. 28(15), 1645–1648 (2016).
[Crossref]

L. Huang, D. Chen, P. Wang, T. Zhang, P. Xiang, Y. Zhang, T. Pu, and X. Chen, “Generation of triangular pulses based on an optoelectronic oscillator,” IEEE Photonics Technol. Lett. 27(23), 2500–2503 (2015).
[Crossref]

Z. Li, H. Chi, X. Zhang, and J. Yao, “Optical single-sideband modulation using a fiber-Bragg-grating-based optical Hilbert transformer,” IEEE Photonics Technol. Lett. 23(9), 558–560 (2011).
[Crossref]

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photonics Technol. Lett. 15(4), 93–96 (2002).

J. Lightwave Technol. (4)

Nat. Photonics (1)

M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
[Crossref]

Opt. Express (2)

Opt. Lett. (5)

Other (1)

A. I. Latkin, S. Boscolo, R. S. Bhamber, and S. K. Turitsyn, “Optical frequency conversion, pulse compression and signal copying using triangular pulses,” in ECOC 2008 (2008), paper Th1B.2.

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Figures (4)

Fig. 1
Fig. 1 (a)Schematic of the proposed triangular microwave waveform generator based on a single PM-DMZM and a polarizer; (b) the working principles. LD: laser diode; PC: polarization controller; DMZM: dual-drive Mach-Zehnder modulator; PM-DMZM: polarization-multiplexing dual-drive Mach-Zehnder modulator; PR: polarization rotator; PBS: polarization beam splitter; PD: photodetector. Pol: polarizer.
Fig. 2
Fig. 2 The optical spectra (a) at the output of the PM-DMZM, (b) and (c) the orthogonally polarized signals split by a PBS connecting the PC following the PM-DMZM.
Fig. 3
Fig. 3 (a) Electrical power spectra of the generated triangular when β1 = 0.25,β2 = 0.71; (b) the eye diagram of the measured triangular waveform and the ideal one.
Fig. 4
Fig. 4 (a) Electrical power spectra of the generated triangular when β1 = 0.44,β2 = 0.53; (b) the eye diagram of the measured triangular waveform and the ideal one.

Equations (9)

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E 1 ( t ) E 0 e j ω 0 t [ e j β 1 cos ( ω m t + θ 1 ) + e j β 1 cos ( ω m t ) e j ϕ 1 ]
E 1 ( t ) E 0 J 0 ( β 1 ) e j ω 0 t [ 1 + e j ϕ 1 ] + j E 0 J 1 ( β 1 ) e j ω 0 t j ω m t e j ϕ 1 [ e j ( θ 1 + ϕ 1 ) + 1 ] + j E 0 J 1 ( β 1 ) e j ω 0 t + j ω m t e j ϕ 1 [ e j ( θ 1 ϕ 1 ) + 1 ]
E 1 ( t ) 2 cos ( ϕ 1 / 2 ) E 0 J 0 ( β 1 ) e j ω 0 t + j ϕ 1 / 2 2 sin ϕ 1 E 0 J 1 ( β 1 ) e j ω 0 t j ω m t
E 2 ( t ) 2 cos ( ϕ 2 / 2 ) E 0 J 0 ( β 2 ) e j ω 0 t + j ϕ 2 / 2 2 sin ϕ 2 E 0 J 1 ( β 2 ) e j ω 0 t + j 3 ω m t + j φ
E ( t ) E 1 ( t ) cos α + E 2 ( t ) sin α
i 0 ( t ) [ J 0 ( β 1 ) J 1 ( β 1 ) cos ( ϕ 1 / 2 ) sin ϕ 1 cos 2 α cos ( ω m t + ϕ 1 / 2 + π ) + J 0 ( β 2 ) J 1 ( β 1 ) cos ( ϕ 2 / 2 ) sin ϕ 1 sin α cos α cos ( ω m t + ϕ 2 / 2 + π ) ] + [ J 0 ( β 1 ) J 1 ( β 2 ) cos ( ϕ 1 / 2 ) sin ϕ 2 sin α cos α cos ( 3 ω m t ϕ 1 / 2 + π + φ ) + J 0 ( β 2 ) J 1 ( β 2 ) cos ( ϕ 2 / 2 ) sin ϕ 2 sin 2 α cos ( 3 ω m t ϕ 2 / 2 + π + φ ) ] + J 1 ( β 1 ) J 1 ( β 2 ) sin ϕ 1 sin ϕ 2 sin α cos α cos ( 4 ω m t + φ )
i 0 ( t ) { [ J 0 ( β 1 ) cos α + J 0 ( β 2 ) sin α ] cos ( ϕ / 2 ) sin ϕ J 1 ( β 1 ) cos α cos ( ω m t + ϕ / 2 + π ) } + { [ J 0 ( β 1 ) cos α + J 0 ( β 2 ) sin α ] cos ( ϕ / 2 ) sin ϕ J 1 ( β 2 ) sin α cos [ 3 ( ω m t + ϕ / 2 + π ) ] } + J 1 ( β 1 ) J 1 ( β 2 ) sin 2 ϕ sin α cos α cos ( 4 ω m t - 2 ϕ )
T ( t ) cos ω t + 1 9 cos 3 ω t + 1 25 cos 5 ω t +
{ [ J 1 ( β 1 ) cos α ] / [ J 1 ( β 2 ) sin α ] = 9 J 1 ( β 1 ) sin ϕ cos α [ J 0 ( β 1 ) cos α + J 0 ( β 2 ) sin α ] cos ( ϕ / 2 )

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