Abstract

Ever since the discovery of space-time duality, several methods have been developed to perform temporal imaging, and there are two major categories: the quadratic signal onto the phase modulator and the parametric mixer with a linear chirped pump. The features of each mechanism have been thoroughly and quantitatively explored and optimized for certain kinds of applications, but a comparison of some key parameters, especially in the aspect of the repetition rate, is required. In this paper, we will first review the theoretical models and existing performance of these two mechanisms and, consequently, compare them quantitatively in different aspects: the focal group delay dispersion, the pupil size, the effective duty ratio, and the temporal numerical aperture. All these fundamental parameters are related to the repetition rate. The results obtained in this study would provide some important guidelines for the time-lens design, so as to be optimized in different kinds of applications with different repetition rate requirements, such as ultrafast optical communication and real-time bio-imaging systems.

© 2013 Optical Society of America

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References

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2013 (2)

C. Zhang, J. Xu, P. C. Chui, and K. K. Y. Wong, “Parametric spectro-temporal analyzer (PASTA) for real-time optical spectrum observation,” Sci. Rep. 3, 2064 (2013).

J. M. Lukens, D. E. Leaird, and A. M. Weiner, “A temporal cloak at telecommunication data rate,” Nature 498, 205–208 (2013).
[CrossRef]

2012 (1)

M. Fridman, A. Farsi, Y. Okawachi, and A. L. Gaeta, “Demonstration of temporal cloaking,” Nature 481, 62–65 (2012).
[CrossRef]

2011 (1)

C. Zhang, K. K. Y. Cheung, P. C. Chui, K. K. Tsia, and K. K. Y. Wong, “Fiber-optical parametric amplifier with high-speed swept pump,” IEEE Photon. Technol. Lett. 23, 1022–1024 (2011).
[CrossRef]

2010 (2)

2009 (4)

2008 (2)

R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Optical time lens based on four-wave mixing on a silicon chip,” Opt. Lett. 33, 1047–1049 (2008).
[CrossRef]

M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456, 81–84 (2008).
[CrossRef]

2007 (1)

2006 (1)

2005 (1)

2004 (3)

J. van Howe, J. Hansryd, and C. Xu, “Multiwavelength pulse generator using time-lens compression,” Opt. Lett. 29, 1470–1472 (2004).
[CrossRef]

J. Azana, N. K. Berger, B. Levit, and B. Fischer, “Spectro-temporal imaging of optical pulses with a single time lens,” IEEE Photon. Technol. Lett. 16, 882–884 (2004).
[CrossRef]

P. J. Almeida, P. Petropoulos, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “All-optical packet compression based on time-to-wavelength conversion,” IEEE Photon. Technol. Lett. 16, 1688–1690 (2004).
[CrossRef]

2003 (1)

2002 (1)

2001 (1)

C. V. Bennett and B. H. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron. 37, 20–32 (2001).
[CrossRef]

2000 (3)

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging—part i: system configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).
[CrossRef]

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging–part ii: system performance,” IEEE J. Quantum Electron. 36, 649–655 (2000).
[CrossRef]

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electro-optic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
[CrossRef]

1999 (2)

1994 (4)

A. A. Godil, B. A. Auld, and D. M. Bloom, “Picosecond time-lenses,” IEEE J. Quantum Electron. 30, 827–837 (1994).
[CrossRef]

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65, 2513–2515 (1994).
[CrossRef]

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

1992 (1)

1989 (1)

1988 (1)

B. H. Kolner, “Active pulse compression using an integrated electro-optic phase modulator,” Appl. Phys. Lett. 52, 1122–1124 (1988).
[CrossRef]

1983 (1)

1981 (1)

1975 (1)

J. E. Bjorkholm, E. H. Turner, and D. B. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near resonant atomic vapor,” Appl. Phys. Lett. 26, 564–566 (1975).
[CrossRef]

1971 (1)

W. J. Caputi, “Stretch: a time-transformation technique,” IEEE Trans. Aerosp. Electron. Syst. AES-7, 269–278 (1971).
[CrossRef]

1969 (1)

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5, 454–458 (1969).
[CrossRef]

1962 (1)

C. F. Buhrer, D. Baird, and E. M. Conwell, “Optical frequency shifting by electro-optic effect,” Appl. Phys. Lett. 1, 46–49 (1962).
[CrossRef]

1960 (1)

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, “The theory and design of chirp radars,” Bell Syst. Tech. J. 39, 745–808 (1960).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

Albersheim, W. J.

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, “The theory and design of chirp radars,” Bell Syst. Tech. J. 39, 745–808 (1960).
[CrossRef]

Almeida, P. J.

P. J. Almeida, P. Petropoulos, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “All-optical packet compression based on time-to-wavelength conversion,” IEEE Photon. Technol. Lett. 16, 1688–1690 (2004).
[CrossRef]

Andres, P.

Atkins, S.

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electro-optic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
[CrossRef]

Auld, B. A.

A. A. Godil, B. A. Auld, and D. M. Bloom, “Picosecond time-lenses,” IEEE J. Quantum Electron. 30, 827–837 (1994).
[CrossRef]

Azana, J.

J. Azana, N. K. Berger, B. Levit, and B. Fischer, “Reconfigurable generation of high-repetition-rate optical pulse sequences based on time-domain phase-only filtering,” Opt. Lett. 30, 3228–3230 (2005).
[CrossRef]

J. Azana, N. K. Berger, B. Levit, and B. Fischer, “Spectro-temporal imaging of optical pulses with a single time lens,” IEEE Photon. Technol. Lett. 16, 882–884 (2004).
[CrossRef]

Baird, D.

C. F. Buhrer, D. Baird, and E. M. Conwell, “Optical frequency shifting by electro-optic effect,” Appl. Phys. Lett. 1, 46–49 (1962).
[CrossRef]

Banyai, W. C.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

Bennett, C. V.

C. V. Bennett and B. H. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron. 37, 20–32 (2001).
[CrossRef]

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging—part i: system configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).
[CrossRef]

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging–part ii: system performance,” IEEE J. Quantum Electron. 36, 649–655 (2000).
[CrossRef]

C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett. 24, 783–785 (1999).
[CrossRef]

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65, 2513–2515 (1994).
[CrossRef]

Berger, N. K.

J. Azana, N. K. Berger, B. Levit, and B. Fischer, “Reconfigurable generation of high-repetition-rate optical pulse sequences based on time-domain phase-only filtering,” Opt. Lett. 30, 3228–3230 (2005).
[CrossRef]

J. Azana, N. K. Berger, B. Levit, and B. Fischer, “Spectro-temporal imaging of optical pulses with a single time lens,” IEEE Photon. Technol. Lett. 16, 882–884 (2004).
[CrossRef]

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electro-optic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
[CrossRef]

Bjorkholm, J. E.

J. E. Bjorkholm, E. H. Turner, and D. B. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near resonant atomic vapor,” Appl. Phys. Lett. 26, 564–566 (1975).
[CrossRef]

Bloom, D. M.

A. A. Godil, B. A. Auld, and D. M. Bloom, “Picosecond time-lenses,” IEEE J. Quantum Electron. 30, 827–837 (1994).
[CrossRef]

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

Bouma, B. E.

Buhrer, C. F.

C. F. Buhrer, D. Baird, and E. M. Conwell, “Optical frequency shifting by electro-optic effect,” Appl. Phys. Lett. 1, 46–49 (1962).
[CrossRef]

Caputi, W. J.

W. J. Caputi, “Stretch: a time-transformation technique,” IEEE Trans. Aerosp. Electron. Syst. AES-7, 269–278 (1971).
[CrossRef]

Cheung, K. K. Y.

C. Zhang, K. K. Y. Cheung, P. C. Chui, K. K. Tsia, and K. K. Y. Wong, “Fiber-optical parametric amplifier with high-speed swept pump,” IEEE Photon. Technol. Lett. 23, 1022–1024 (2011).
[CrossRef]

Chui, P. C.

C. Zhang, J. Xu, P. C. Chui, and K. K. Y. Wong, “Parametric spectro-temporal analyzer (PASTA) for real-time optical spectrum observation,” Sci. Rep. 3, 2064 (2013).

C. Zhang, K. K. Y. Cheung, P. C. Chui, K. K. Tsia, and K. K. Y. Wong, “Fiber-optical parametric amplifier with high-speed swept pump,” IEEE Photon. Technol. Lett. 23, 1022–1024 (2011).
[CrossRef]

Conwell, E. M.

C. F. Buhrer, D. Baird, and E. M. Conwell, “Optical frequency shifting by electro-optic effect,” Appl. Phys. Lett. 1, 46–49 (1962).
[CrossRef]

Corsini, R.

Darlington, S.

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, “The theory and design of chirp radars,” Bell Syst. Tech. J. 39, 745–808 (1960).
[CrossRef]

de Boer, J. F.

Farsi, A.

M. Fridman, A. Farsi, Y. Okawachi, and A. L. Gaeta, “Demonstration of temporal cloaking,” Nature 481, 62–65 (2012).
[CrossRef]

Fischer, B.

J. Azana, N. K. Berger, B. Levit, and B. Fischer, “Reconfigurable generation of high-repetition-rate optical pulse sequences based on time-domain phase-only filtering,” Opt. Lett. 30, 3228–3230 (2005).
[CrossRef]

J. Azana, N. K. Berger, B. Levit, and B. Fischer, “Spectro-temporal imaging of optical pulses with a single time lens,” IEEE Photon. Technol. Lett. 16, 882–884 (2004).
[CrossRef]

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electro-optic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
[CrossRef]

Foster, M. A.

Franco, P.

Fridman, M.

M. Fridman, A. Farsi, Y. Okawachi, and A. L. Gaeta, “Demonstration of temporal cloaking,” Nature 481, 62–65 (2012).
[CrossRef]

Gaeta, A. L.

Geraghty, D. F.

Godil, A. A.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

A. A. Godil, B. A. Auld, and D. M. Bloom, “Picosecond time-lenses,” IEEE J. Quantum Electron. 30, 827–837 (1994).
[CrossRef]

Hagimoto, K.

Hansryd, J.

Ibsen, M.

P. J. Almeida, P. Petropoulos, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “All-optical packet compression based on time-to-wavelength conversion,” IEEE Photon. Technol. Lett. 16, 1688–1690 (2004).
[CrossRef]

Iftimia, N.

Jannson, J.

Jannson, T.

Kauffman, M. T.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–272 (1994).
[CrossRef]

Klauder, J. R.

J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, “The theory and design of chirp radars,” Bell Syst. Tech. J. 39, 745–808 (1960).
[CrossRef]

Kolner, B. H.

C. V. Bennett and B. H. Kolner, “Aberrations in temporal imaging,” IEEE J. Quantum Electron. 37, 20–32 (2001).
[CrossRef]

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging–part ii: system performance,” IEEE J. Quantum Electron. 36, 649–655 (2000).
[CrossRef]

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging—part i: system configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).
[CrossRef]

C. V. Bennett and B. H. Kolner, “Upconversion time microscope demonstrating 103× magnification of femtosecond waveforms,” Opt. Lett. 24, 783–785 (1999).
[CrossRef]

C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65, 2513–2515 (1994).
[CrossRef]

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30, 1951–1963 (1994).
[CrossRef]

B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14, 630–632 (1989).
[CrossRef]

B. H. Kolner, “Active pulse compression using an integrated electro-optic phase modulator,” Appl. Phys. Lett. 52, 1122–1124 (1988).
[CrossRef]

B. H. Kolner, “Electro-optic time lenses for shaping and imaging optical waveforms,” in Modulators for Optical Communications, Science, Technology, and Applications, A. Chen and E. Murphy, eds. (CRC Press, 2012), pp. 427–454.

Kuzucu, O.

Lancis, J.

Leaird, D. E.

Lee, J. H.

Levit, B.

J. Azana, N. K. Berger, B. Levit, and B. Fischer, “Reconfigurable generation of high-repetition-rate optical pulse sequences based on time-domain phase-only filtering,” Opt. Lett. 30, 3228–3230 (2005).
[CrossRef]

J. Azana, N. K. Berger, B. Levit, and B. Fischer, “Spectro-temporal imaging of optical pulses with a single time lens,” IEEE Photon. Technol. Lett. 16, 882–884 (2004).
[CrossRef]

N. K. Berger, B. Levit, S. Atkins, and B. Fischer, “Time-lens-based spectral analysis of optical pulses by electro-optic phase modulation,” Electron. Lett. 36, 1644–1646 (2000).
[CrossRef]

Lipson, M.

Lohmann, A. W.

Long, C. M.

Lukens, J. M.

J. M. Lukens, D. E. Leaird, and A. M. Weiner, “A temporal cloak at telecommunication data rate,” Nature 498, 205–208 (2013).
[CrossRef]

Marhic, M. E.

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge University, 2007).

Mendlovic, D.

Midrio, M.

Morioka, T.

Munioz-Camuniez, L. E.

Nazarathy, M.

Ojeda-Castaneda, J.

Okawachi, Y.

M. Fridman, A. Farsi, Y. Okawachi, and A. L. Gaeta, “Demonstration of temporal cloaking,” Nature 481, 62–65 (2012).
[CrossRef]

O. Kuzucu, Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Spectral phase conjugation via temporal imaging,” Opt. Express 17, 20605–20614 (2009).
[CrossRef]

Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “High-resolution spectroscopy using a frequency magnifier,” Opt. Express 17, 5691–5697 (2009).
[CrossRef]

M. A. Foster, R. Salem, Y. Okawachi, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Ultrafast waveform compression using a time-domain telescope,” Nat. Photonics 3, 581–585 (2009).
[CrossRef]

Pawley, J. B.

J. B. Pawley, Handbook of Biological Confocal Microscopy, 3rd ed. (Springer Berlin, 2006).

Pearson, D. B.

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Figures (8)

Fig. 1.
Fig. 1.

Relation between the focal GDD Φf and the repetition rate ft for the time-lens applications based on electro-optic phase modulator and parametric mixer. The numbers inside each spot refer to the corresponding number in references section.

Fig. 2.
Fig. 2.

Schematic diagram of the time-lens. The transformation flow between the time and frequency domain is followed with the arrow directions.

Fig. 3.
Fig. 3.

Principle of the phase modulator is to realize the linear frequency sweep and act as the time-lens: (a) ideal quadratic signal applied to the phase modulator, and linear frequency sweep has been achieved; (b) the cusp of the sinusoidal signal approximates to the quadratic signal, and here the frequency trace is the derivation of phase shift. The error ratio η is defined in the frequency domain: the deviation from the real frequency a (solid line) to the ideal frequency b (dashed line), over the ideal frequency b, namely η=(ba)/b.

Fig. 4.
Fig. 4.

Principle of the parametric mixer based on FWM to realize the linear frequency sweep and act as the time-lens. (a) Generation of the chirped pump. (b) Frequency transformation during the FWM process.

Fig. 5.
Fig. 5.

Effect of repetition rate on the focal GDD range for two time-lens mechanisms. Solid line: constraints on the phase modulator (1) and (2). Dashed–dotted line: constraints on the parametric mixer (3) and (4). Two sets of dots correspond to the relations shown in Fig. 1.

Fig. 6.
Fig. 6.

(a) Pupil size versus the focal GDD: constraints on the phase modulator (red-shaded area enclosed by solid line), and constraints on the parametric mixer (blue-shaded area enclosed by dashed–dotted lines). Two sets of dots correspond to the relations shown in Fig. 1. (b) The effective duty ratio of two time-lens mechanisms, phase modulator (red solid: sinusoidal, blue dash: Gaussian); parametric mixer (dots).

Fig. 7.
Fig. 7.

Effect of repetition rate on the temporal NA performance for two time-lens mechanisms. Solid line: constraints on the phase modulator (1) and (2). Dashed–dotted line: constraints on the parametric mixer (3) and (4). Two sets of dots correspond to the relations shown in Fig. 1.

Fig. 8.
Fig. 8.

Design guidelines of two time-lens mechanisms. The red-shaded area on the top-left corner corresponds to the phase modulator, and the blue-shaded area on the bottom-right corner corresponds to the parametric mixer. The dashed–dotted lines refer to the NAt value, and the direction of arrows reflects the increase/decrease of the corresponding parameters.

Equations (26)

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a(t)=I1{A(ω)Gi(ω)},
tf(t)=exp(it22Φf).
e(t)=I1{[A(ω)Gi(ω)*Tf(ω)]Go(ω)}=12π+dωexp(iωt)Go(ω)·+dωA(ω)Gi(ω)Tf(ωω)=+dωA(ω)Gi(ω)exp[iΦf2ω2]·2πiΦf2π+exp[iω(tΦfω)]exp[iΦfΦo2ω2]dω.
e(t)=ΦfΦfΦo+dωA(ω)Gi(ω)×exp[iΦf2ω2]exp[i(tΦfω)22(ΦfΦo)]=ΦfΦfΦoexp[it22(ΦfΦo)]×+dωA(ω)exp[iΦfωΦfΦot]×exp[i(1Φo+1Φi1Φf)ΦiΦoΦf2(ΦfΦo)ω2].
e(t)=2πΦiΦoexp[iΦi2ΦoΦft2]a(ΦiΦot).
e(t)=+dωA(ω)Gi(ω)exp[iΦf2ω2]·2πiΦf2π+exp[iω(tΦfω)]dω=2πiΦfA(tΦf).
ϕ(t)=Aϕ[cos(ωmt)1]t22Φf,
Φf_PM=1ϕ(t)1ϕ(0)=1Aϕωm2.
apo(t)=Apo(t)exp(iϕpo)=exp[2ln2(ttPW)2+iωpot],
ap(t)exp[2ln21+iε(ttPW)2]exp[iωpot1+iε]=apo11+iε(t).
ap(t)=Ap(t)exp[iϕp(t)]=exp[(ttPW)22ln21+ε2+ωpotε1+ε2]×exp[i(ttPW)22εln21+ε2+iωpot1+ε2].
T3dB=tPW1+ε2.
ωp(t)=ϕp(t+tc)t=11+ε2[4ln2εtPW2t+4ln2εtPW2Φpωpo+ωpo]=11+ε2[ε2Φpt+(1+ε2)ωpo]=t(1+ε2)Φp+ωpo.
ωi(t)=2ωp(t)ωs(t)=2ωpoωs(t)+2tΦp(1+ε2).
ω(t)=[ωi(t)ωs(t)]t=2Φp(1+ε2)2Φp.
Φf_FWM=1ϕ(t)=1ω(t)Φp2.
Φf_PM=1Aϕωm2>1(2πft)2Aϕmax.
Δλ=λ022πcΔω=λ022πc|ωp(t)t|Tt=λ024πcftΦf_FWM.
tPW28ln2=ΦRp2<Φf_FWM<λ024πcftΔλmin.
ω(t)=tΦf_PM(1η)=Aϕωmsin(ωmt).
ΔTPM=2πsinc1(1η)ωm,
Rt_FWM<(log2)12n.
NAt_PM=ΔTPM2Φf<2π2Aϕsinc1(1η)ftmax.
δtPM=ln2π2Aϕsinc1(1η)ft,
NAt_FWM=T3dB2Φf=4ln2tPW1+ε24ln2tPW.
δtFWM=4ln2Δωf=2ln2λ02πcΔλ.

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