Abstract

We propose a novel scheme of temporal Talbot effect achieving optical pulse train repetition-rate multiplication in a conventional tapped delay line structure. While it is generally used for spectral amplitude filtering, we show that such architecture could also be configured for spectral phase-only filtering, as well as for a combination of amplitude and phase filtering regimes. We theoretically derive and numerically simulate the working principle of the concept, followed by a proof-of-principle experimental demonstration using an off-the-shelf Mach-Zehnder delay line interferometer, which corresponds to the simplest version of the proposed structure. We address the efficiency, and potential performance degradation in the presence of power imbalance and delay line length inaccuracy of the architecture, together with applied phase error.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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  1. M. Nakazawa, E. Yoshida, T. Yamamoto, E. Yamada, and A. Sahara, “TDM single channel 640Gbit/s transmission experiment over 60 km using 400fs pulse train and walk-off free, dispersion flattened nonlinear optical loop mirror,” Electron. Lett. 34(9), 907–908 (1998).
    [Crossref]
  2. F.-M. Kuo, C.-B. Huang, J.-W. Shi, N.-W. Chen, H.-P. Chuang, J. E. Bowers, and C.-L. Pan, “Remotely up-converted 20-Gbit/s error-free wireless on-off-keying data transmission at W-band using an ultra-wideband photonic transmitter-mixer,” IEEE Photonics J. 3(2), 209–219 (2011).
    [Crossref]
  3. V. Torres-Company and A. M. Weiner, “Optical frequency comb technology for ultra-broadband radiofrequency photonics,” Laser Photonics Rev. 8(3), 368–393 (2014).
    [Crossref]
  4. J.-M. Wun, H.-Y. Liu, Y.-L. Zeng, S.-D. Yang, C.-L. Pan, C.-B. Huang, and J.-W. Shi, “Photonic High-Power Continuous Wave THz-Wave Generation by Using Flip-Chip Packaged Uni-Traveling Carrier Photodiodes and a Femtosecond Optical Pulse Generator,” J. Lightwave Technol. 34(4), 1387–1397 (2016).
    [Crossref]
  5. J. P. Yao, “Photonic generation of microwave arbitrary waveforms,” Opt. Commun. 284(15), 3723–3736 (2011).
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  6. V. Torres-Company, A. J. Metcalf, D. Leaird, and A. M. Weiner, “Multichannel radio-frequency arbitrary waveform generation based on multiwavelength comb switching and 2-D line-by-line pulse shaping,” IEEE Photonics Technol. Lett. 24(11), 891–893 (2012).
    [Crossref]
  7. V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6(3), 186–194 (2012).
    [Crossref]
  8. R. Maram, L. R. Cortés, and J. Azaña, “Programmable fiber-optics pulse repetition-rate multiplier,” J. Lightwave Technol. 34(2), 448–455 (2016).
    [Crossref]
  9. Y. Xie, L. Zhuang, and A. J. Lowery, “Picosecond optical pulse processing using a terahertz-bandwidth reconfigurable photonic integrated circuit,” Nanophotonics 7(5), 837–852 (2018).
    [Crossref]
  10. D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010).
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  11. Z. Geng, Y. Xie, L. Zhuang, M. Burla, M. Hoekman, C. G. H. Roeloffzen, and A. J. Lowery, “Photonic integrated circuit implementation of a sub-GHz-selectivity frequency comb filter for optical clock multiplication,” Opt. Express 25(22), 27635–27645 (2017).
    [Crossref] [PubMed]
  12. J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7(4), 728–744 (2001).
    [Crossref]
  13. J. Caraquitena, Z. Jiang, D. E. Leaird, and A. M. Weiner, “Tunable pulse repetition-rate multiplication using phase-only line-by-line pulse shaping,” Opt. Lett. 32(6), 716–718 (2007).
    [Crossref] [PubMed]
  14. C.-C. Chen, I.-C. Hsieh, S.-D. Yang, and C.-B. Huang, “Polarization line-by-line pulse shaping for the implementation of vectorial temporal Talbot effect,” Opt. Express 20(24), 27062–27070 (2012).
    [Crossref] [PubMed]
  15. C.-B. Huang and Y. Lai, “Loss-less pulse intensity repetition-rate multiplication using optical all-pass filtering,” IEEE Photonics Technol. Lett. 12(2), 167–169 (2000).
    [Crossref]
  16. M. A. Preciado and M. A. Muriel, “All-pass optical structures for repetition rate multiplication,” Opt. Express 16(15), 11162–11168 (2008).
    [Crossref] [PubMed]
  17. H.-P. Chuang and C.-B. Huang, “Generation and delivery of 1-ps optical pulses with ultrahigh repetition-rates over 25-km single mode fiber by a spectral line-by-line pulse shaper,” Opt. Express 18(23), 24003–24011 (2010).
    [Crossref] [PubMed]
  18. M. Seghilani, R. Maram, and J. Azaña, “Mitigating nonlinear propagation impairments of ultrashort pulses by fractional temporal self-imaging,” Opt. Lett. 42(4), 879–882 (2017).
    [Crossref] [PubMed]
  19. R. Maram, J. Van Howe, M. Li, and J. Azaña, “Noiseless intensity amplification of repetitive signals by coherent addition using the temporal Talbot effect,” Nat. Commun. 5(1), 5163 (2014).
    [Crossref] [PubMed]
  20. L. R. Cortés, M. Seghilani, R. Maram, and J. Azaña, “Full-field broadband invisibility through reversible wave frequency-spectrum control,” Optica 5(7), 779–786 (2018).
    [Crossref]
  21. M. A. Soto, M. Alem, M. Amin Shoaie, A. Vedadi, C. S. Brès, L. Thévenaz, and T. Schneider, “Optical sinc-shaped Nyquist pulses of exceptional quality,” Nat. Commun. 4(1), 2898 (2013).
    [Crossref] [PubMed]
  22. S. Liao, Y. Ding, J. Dong, T. Yang, X. Chen, D. Gao, and X. Zhang, “Arbitrary waveform generator and differentiator employing an integrated optical pulse shaper,” Opt. Express 23(9), 12161–12173 (2015).
    [Crossref] [PubMed]
  23. S. Min, Y. Zhao, and S. Fleming, “Repetition rate multiplication in figure-eight fibre laser with 3 dB couplers,” Opt. Commun. 277(2), 411–413 (2007).
    [Crossref]
  24. L. R. Cortés, H. Guillet de Chatellus, and J. Azaña, “On the generality of the Talbot condition for inducing self-imaging effects on periodic objects,” Opt. Lett. 41(2), 340–343 (2016).
    [Crossref] [PubMed]
  25. C. R. Fernández-Pousa, “On the structure of quadratic Gauss sums in the Talbot effect,” J. Opt. Soc. Am. A 34(5), 732–742 (2017).
    [Crossref] [PubMed]
  26. C. R. Fernández-Pousa, R. Maram, and J. Azaña, “CW-to-pulse conversion using temporal Talbot array illuminators,” Opt. Lett. 42(13), 2427–2430 (2017).
    [Crossref] [PubMed]
  27. M. Bachmann, P. A. Besse, and H. Melchior, “General self-imaging properties in N × N multimode interference couplers including phase relations,” Appl. Opt. 33(18), 3905–3911 (1994).
    [Crossref] [PubMed]
  28. J. Caraquitena, M. Beltrán, R. Llorente, J. Martí, and M. A. Muriel, “Spectral self-imaging effect by time-domain multilevel phase modulation of a periodic pulse train,” Opt. Lett. 36(6), 858–860 (2011).
    [Crossref] [PubMed]
  29. A. Malacarne and J. Azaña, “Discretely tunable comb spacing of a frequency comb by multilevel phase modulation of a periodic pulse train,” Opt. Express 21(4), 4139–4144 (2013).
    [Crossref] [PubMed]
  30. J. Hu, S. Fabbri, and C.-S. Brès, “Flexible width Nyquist pulse based on a single Mach-Zehnder modulator,” in CLEO: Science and Innovations (Optical Society of America, 2018), paper SF3N.6.
    [Crossref]
  31. M. E. Marhic, “Discrete Fourier transforms by single-mode star networks,” Opt. Lett. 12(1), 63–65 (1987).
    [Crossref] [PubMed]
  32. K. Wörhoff, R. G. Heideman, A. Leinse, and M. Hoekman, “TriPleX: a versatile dielectric photonic platform,” Adv. Opt. Technol. 4(2), 189–207 (2015).
  33. Y. Li, J. Li, H. Yu, H. Yu, H. Chen, S. Yang, and M. Chen, “On-chip photonic microsystem for optical signal processing based on silicon and silicon nitride platforms,” Adv. Opt. Technol. 7(1-2), 81–101 (2018).
    [Crossref]
  34. Y. Dai and J. Yao, “Nonuniformly-spaced photonic microwave delayline filter,” Opt. Express 16(7), 4713–4718 (2008).
    [Crossref] [PubMed]
  35. B. C. Berndt and R. J. Evans, “The determination of Gauss sums,” Bull. Am. Math. Soc. (N.S.) 5(2), 107–129 (1981).
    [Crossref]

2018 (3)

Y. Xie, L. Zhuang, and A. J. Lowery, “Picosecond optical pulse processing using a terahertz-bandwidth reconfigurable photonic integrated circuit,” Nanophotonics 7(5), 837–852 (2018).
[Crossref]

L. R. Cortés, M. Seghilani, R. Maram, and J. Azaña, “Full-field broadband invisibility through reversible wave frequency-spectrum control,” Optica 5(7), 779–786 (2018).
[Crossref]

Y. Li, J. Li, H. Yu, H. Yu, H. Chen, S. Yang, and M. Chen, “On-chip photonic microsystem for optical signal processing based on silicon and silicon nitride platforms,” Adv. Opt. Technol. 7(1-2), 81–101 (2018).
[Crossref]

2017 (4)

C. R. Fernández-Pousa, “On the structure of quadratic Gauss sums in the Talbot effect,” J. Opt. Soc. Am. A 34(5), 732–742 (2017).
[Crossref] [PubMed]

C. R. Fernández-Pousa, R. Maram, and J. Azaña, “CW-to-pulse conversion using temporal Talbot array illuminators,” Opt. Lett. 42(13), 2427–2430 (2017).
[Crossref] [PubMed]

M. Seghilani, R. Maram, and J. Azaña, “Mitigating nonlinear propagation impairments of ultrashort pulses by fractional temporal self-imaging,” Opt. Lett. 42(4), 879–882 (2017).
[Crossref] [PubMed]

Z. Geng, Y. Xie, L. Zhuang, M. Burla, M. Hoekman, C. G. H. Roeloffzen, and A. J. Lowery, “Photonic integrated circuit implementation of a sub-GHz-selectivity frequency comb filter for optical clock multiplication,” Opt. Express 25(22), 27635–27645 (2017).
[Crossref] [PubMed]

2016 (3)

J.-M. Wun, H.-Y. Liu, Y.-L. Zeng, S.-D. Yang, C.-L. Pan, C.-B. Huang, and J.-W. Shi, “Photonic High-Power Continuous Wave THz-Wave Generation by Using Flip-Chip Packaged Uni-Traveling Carrier Photodiodes and a Femtosecond Optical Pulse Generator,” J. Lightwave Technol. 34(4), 1387–1397 (2016).
[Crossref]

R. Maram, L. R. Cortés, and J. Azaña, “Programmable fiber-optics pulse repetition-rate multiplier,” J. Lightwave Technol. 34(2), 448–455 (2016).
[Crossref]

L. R. Cortés, H. Guillet de Chatellus, and J. Azaña, “On the generality of the Talbot condition for inducing self-imaging effects on periodic objects,” Opt. Lett. 41(2), 340–343 (2016).
[Crossref] [PubMed]

2015 (2)

S. Liao, Y. Ding, J. Dong, T. Yang, X. Chen, D. Gao, and X. Zhang, “Arbitrary waveform generator and differentiator employing an integrated optical pulse shaper,” Opt. Express 23(9), 12161–12173 (2015).
[Crossref] [PubMed]

K. Wörhoff, R. G. Heideman, A. Leinse, and M. Hoekman, “TriPleX: a versatile dielectric photonic platform,” Adv. Opt. Technol. 4(2), 189–207 (2015).

2014 (2)

R. Maram, J. Van Howe, M. Li, and J. Azaña, “Noiseless intensity amplification of repetitive signals by coherent addition using the temporal Talbot effect,” Nat. Commun. 5(1), 5163 (2014).
[Crossref] [PubMed]

V. Torres-Company and A. M. Weiner, “Optical frequency comb technology for ultra-broadband radiofrequency photonics,” Laser Photonics Rev. 8(3), 368–393 (2014).
[Crossref]

2013 (2)

M. A. Soto, M. Alem, M. Amin Shoaie, A. Vedadi, C. S. Brès, L. Thévenaz, and T. Schneider, “Optical sinc-shaped Nyquist pulses of exceptional quality,” Nat. Commun. 4(1), 2898 (2013).
[Crossref] [PubMed]

A. Malacarne and J. Azaña, “Discretely tunable comb spacing of a frequency comb by multilevel phase modulation of a periodic pulse train,” Opt. Express 21(4), 4139–4144 (2013).
[Crossref] [PubMed]

2012 (3)

V. Torres-Company, A. J. Metcalf, D. Leaird, and A. M. Weiner, “Multichannel radio-frequency arbitrary waveform generation based on multiwavelength comb switching and 2-D line-by-line pulse shaping,” IEEE Photonics Technol. Lett. 24(11), 891–893 (2012).
[Crossref]

V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6(3), 186–194 (2012).
[Crossref]

C.-C. Chen, I.-C. Hsieh, S.-D. Yang, and C.-B. Huang, “Polarization line-by-line pulse shaping for the implementation of vectorial temporal Talbot effect,” Opt. Express 20(24), 27062–27070 (2012).
[Crossref] [PubMed]

2011 (3)

J. P. Yao, “Photonic generation of microwave arbitrary waveforms,” Opt. Commun. 284(15), 3723–3736 (2011).
[Crossref]

F.-M. Kuo, C.-B. Huang, J.-W. Shi, N.-W. Chen, H.-P. Chuang, J. E. Bowers, and C.-L. Pan, “Remotely up-converted 20-Gbit/s error-free wireless on-off-keying data transmission at W-band using an ultra-wideband photonic transmitter-mixer,” IEEE Photonics J. 3(2), 209–219 (2011).
[Crossref]

J. Caraquitena, M. Beltrán, R. Llorente, J. Martí, and M. A. Muriel, “Spectral self-imaging effect by time-domain multilevel phase modulation of a periodic pulse train,” Opt. Lett. 36(6), 858–860 (2011).
[Crossref] [PubMed]

2010 (2)

H.-P. Chuang and C.-B. Huang, “Generation and delivery of 1-ps optical pulses with ultrahigh repetition-rates over 25-km single mode fiber by a spectral line-by-line pulse shaper,” Opt. Express 18(23), 24003–24011 (2010).
[Crossref] [PubMed]

D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010).
[Crossref] [PubMed]

2008 (2)

M. A. Preciado and M. A. Muriel, “All-pass optical structures for repetition rate multiplication,” Opt. Express 16(15), 11162–11168 (2008).
[Crossref] [PubMed]

Y. Dai and J. Yao, “Nonuniformly-spaced photonic microwave delayline filter,” Opt. Express 16(7), 4713–4718 (2008).
[Crossref] [PubMed]

2007 (2)

S. Min, Y. Zhao, and S. Fleming, “Repetition rate multiplication in figure-eight fibre laser with 3 dB couplers,” Opt. Commun. 277(2), 411–413 (2007).
[Crossref]

J. Caraquitena, Z. Jiang, D. E. Leaird, and A. M. Weiner, “Tunable pulse repetition-rate multiplication using phase-only line-by-line pulse shaping,” Opt. Lett. 32(6), 716–718 (2007).
[Crossref] [PubMed]

2001 (1)

J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7(4), 728–744 (2001).
[Crossref]

2000 (1)

C.-B. Huang and Y. Lai, “Loss-less pulse intensity repetition-rate multiplication using optical all-pass filtering,” IEEE Photonics Technol. Lett. 12(2), 167–169 (2000).
[Crossref]

1998 (1)

M. Nakazawa, E. Yoshida, T. Yamamoto, E. Yamada, and A. Sahara, “TDM single channel 640Gbit/s transmission experiment over 60 km using 400fs pulse train and walk-off free, dispersion flattened nonlinear optical loop mirror,” Electron. Lett. 34(9), 907–908 (1998).
[Crossref]

1994 (1)

M. Bachmann, P. A. Besse, and H. Melchior, “General self-imaging properties in N × N multimode interference couplers including phase relations,” Appl. Opt. 33(18), 3905–3911 (1994).
[Crossref] [PubMed]

1987 (1)

M. E. Marhic, “Discrete Fourier transforms by single-mode star networks,” Opt. Lett. 12(1), 63–65 (1987).
[Crossref] [PubMed]

1981 (1)

B. C. Berndt and R. J. Evans, “The determination of Gauss sums,” Bull. Am. Math. Soc. (N.S.) 5(2), 107–129 (1981).
[Crossref]

Alem, M.

M. A. Soto, M. Alem, M. Amin Shoaie, A. Vedadi, C. S. Brès, L. Thévenaz, and T. Schneider, “Optical sinc-shaped Nyquist pulses of exceptional quality,” Nat. Commun. 4(1), 2898 (2013).
[Crossref] [PubMed]

Amin Shoaie, M.

M. A. Soto, M. Alem, M. Amin Shoaie, A. Vedadi, C. S. Brès, L. Thévenaz, and T. Schneider, “Optical sinc-shaped Nyquist pulses of exceptional quality,” Nat. Commun. 4(1), 2898 (2013).
[Crossref] [PubMed]

Azaña, J.

L. R. Cortés, M. Seghilani, R. Maram, and J. Azaña, “Full-field broadband invisibility through reversible wave frequency-spectrum control,” Optica 5(7), 779–786 (2018).
[Crossref]

M. Seghilani, R. Maram, and J. Azaña, “Mitigating nonlinear propagation impairments of ultrashort pulses by fractional temporal self-imaging,” Opt. Lett. 42(4), 879–882 (2017).
[Crossref] [PubMed]

C. R. Fernández-Pousa, R. Maram, and J. Azaña, “CW-to-pulse conversion using temporal Talbot array illuminators,” Opt. Lett. 42(13), 2427–2430 (2017).
[Crossref] [PubMed]

L. R. Cortés, H. Guillet de Chatellus, and J. Azaña, “On the generality of the Talbot condition for inducing self-imaging effects on periodic objects,” Opt. Lett. 41(2), 340–343 (2016).
[Crossref] [PubMed]

R. Maram, L. R. Cortés, and J. Azaña, “Programmable fiber-optics pulse repetition-rate multiplier,” J. Lightwave Technol. 34(2), 448–455 (2016).
[Crossref]

R. Maram, J. Van Howe, M. Li, and J. Azaña, “Noiseless intensity amplification of repetitive signals by coherent addition using the temporal Talbot effect,” Nat. Commun. 5(1), 5163 (2014).
[Crossref] [PubMed]

A. Malacarne and J. Azaña, “Discretely tunable comb spacing of a frequency comb by multilevel phase modulation of a periodic pulse train,” Opt. Express 21(4), 4139–4144 (2013).
[Crossref] [PubMed]

J. Azaña and M. A. Muriel, “Temporal self-imaging effects: theory and application for multiplying pulse repetition rates,” IEEE J. Sel. Top. Quantum Electron. 7(4), 728–744 (2001).
[Crossref]

Bachmann, M.

M. Bachmann, P. A. Besse, and H. Melchior, “General self-imaging properties in N × N multimode interference couplers including phase relations,” Appl. Opt. 33(18), 3905–3911 (1994).
[Crossref] [PubMed]

Beltrán, M.

J. Caraquitena, M. Beltrán, R. Llorente, J. Martí, and M. A. Muriel, “Spectral self-imaging effect by time-domain multilevel phase modulation of a periodic pulse train,” Opt. Lett. 36(6), 858–860 (2011).
[Crossref] [PubMed]

Ben Ezra, S.

D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010).
[Crossref] [PubMed]

Berndt, B. C.

B. C. Berndt and R. J. Evans, “The determination of Gauss sums,” Bull. Am. Math. Soc. (N.S.) 5(2), 107–129 (1981).
[Crossref]

Besse, P. A.

M. Bachmann, P. A. Besse, and H. Melchior, “General self-imaging properties in N × N multimode interference couplers including phase relations,” Appl. Opt. 33(18), 3905–3911 (1994).
[Crossref] [PubMed]

Bowers, J. E.

F.-M. Kuo, C.-B. Huang, J.-W. Shi, N.-W. Chen, H.-P. Chuang, J. E. Bowers, and C.-L. Pan, “Remotely up-converted 20-Gbit/s error-free wireless on-off-keying data transmission at W-band using an ultra-wideband photonic transmitter-mixer,” IEEE Photonics J. 3(2), 209–219 (2011).
[Crossref]

Brès, C. S.

M. A. Soto, M. Alem, M. Amin Shoaie, A. Vedadi, C. S. Brès, L. Thévenaz, and T. Schneider, “Optical sinc-shaped Nyquist pulses of exceptional quality,” Nat. Commun. 4(1), 2898 (2013).
[Crossref] [PubMed]

Burla, M.

Z. Geng, Y. Xie, L. Zhuang, M. Burla, M. Hoekman, C. G. H. Roeloffzen, and A. J. Lowery, “Photonic integrated circuit implementation of a sub-GHz-selectivity frequency comb filter for optical clock multiplication,” Opt. Express 25(22), 27635–27645 (2017).
[Crossref] [PubMed]

Caraquitena, J.

J. Caraquitena, M. Beltrán, R. Llorente, J. Martí, and M. A. Muriel, “Spectral self-imaging effect by time-domain multilevel phase modulation of a periodic pulse train,” Opt. Lett. 36(6), 858–860 (2011).
[Crossref] [PubMed]

J. Caraquitena, Z. Jiang, D. E. Leaird, and A. M. Weiner, “Tunable pulse repetition-rate multiplication using phase-only line-by-line pulse shaping,” Opt. Lett. 32(6), 716–718 (2007).
[Crossref] [PubMed]

Chen, C.-C.

C.-C. Chen, I.-C. Hsieh, S.-D. Yang, and C.-B. Huang, “Polarization line-by-line pulse shaping for the implementation of vectorial temporal Talbot effect,” Opt. Express 20(24), 27062–27070 (2012).
[Crossref] [PubMed]

Chen, H.

Y. Li, J. Li, H. Yu, H. Yu, H. Chen, S. Yang, and M. Chen, “On-chip photonic microsystem for optical signal processing based on silicon and silicon nitride platforms,” Adv. Opt. Technol. 7(1-2), 81–101 (2018).
[Crossref]

Chen, M.

Y. Li, J. Li, H. Yu, H. Yu, H. Chen, S. Yang, and M. Chen, “On-chip photonic microsystem for optical signal processing based on silicon and silicon nitride platforms,” Adv. Opt. Technol. 7(1-2), 81–101 (2018).
[Crossref]

Chen, N.-W.

F.-M. Kuo, C.-B. Huang, J.-W. Shi, N.-W. Chen, H.-P. Chuang, J. E. Bowers, and C.-L. Pan, “Remotely up-converted 20-Gbit/s error-free wireless on-off-keying data transmission at W-band using an ultra-wideband photonic transmitter-mixer,” IEEE Photonics J. 3(2), 209–219 (2011).
[Crossref]

Chen, X.

S. Liao, Y. Ding, J. Dong, T. Yang, X. Chen, D. Gao, and X. Zhang, “Arbitrary waveform generator and differentiator employing an integrated optical pulse shaper,” Opt. Express 23(9), 12161–12173 (2015).
[Crossref] [PubMed]

Chuang, H.-P.

F.-M. Kuo, C.-B. Huang, J.-W. Shi, N.-W. Chen, H.-P. Chuang, J. E. Bowers, and C.-L. Pan, “Remotely up-converted 20-Gbit/s error-free wireless on-off-keying data transmission at W-band using an ultra-wideband photonic transmitter-mixer,” IEEE Photonics J. 3(2), 209–219 (2011).
[Crossref]

H.-P. Chuang and C.-B. Huang, “Generation and delivery of 1-ps optical pulses with ultrahigh repetition-rates over 25-km single mode fiber by a spectral line-by-line pulse shaper,” Opt. Express 18(23), 24003–24011 (2010).
[Crossref] [PubMed]

Cortés, L. R.

L. R. Cortés, M. Seghilani, R. Maram, and J. Azaña, “Full-field broadband invisibility through reversible wave frequency-spectrum control,” Optica 5(7), 779–786 (2018).
[Crossref]

L. R. Cortés, H. Guillet de Chatellus, and J. Azaña, “On the generality of the Talbot condition for inducing self-imaging effects on periodic objects,” Opt. Lett. 41(2), 340–343 (2016).
[Crossref] [PubMed]

R. Maram, L. R. Cortés, and J. Azaña, “Programmable fiber-optics pulse repetition-rate multiplier,” J. Lightwave Technol. 34(2), 448–455 (2016).
[Crossref]

Dai, Y.

Y. Dai and J. Yao, “Nonuniformly-spaced photonic microwave delayline filter,” Opt. Express 16(7), 4713–4718 (2008).
[Crossref] [PubMed]

Ding, Y.

S. Liao, Y. Ding, J. Dong, T. Yang, X. Chen, D. Gao, and X. Zhang, “Arbitrary waveform generator and differentiator employing an integrated optical pulse shaper,” Opt. Express 23(9), 12161–12173 (2015).
[Crossref] [PubMed]

Dong, J.

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K. Wörhoff, R. G. Heideman, A. Leinse, and M. Hoekman, “TriPleX: a versatile dielectric photonic platform,” Adv. Opt. Technol. 4(2), 189–207 (2015).

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D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010).
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C.-C. Chen, I.-C. Hsieh, S.-D. Yang, and C.-B. Huang, “Polarization line-by-line pulse shaping for the implementation of vectorial temporal Talbot effect,” Opt. Express 20(24), 27062–27070 (2012).
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Huang, C.-B.

J.-M. Wun, H.-Y. Liu, Y.-L. Zeng, S.-D. Yang, C.-L. Pan, C.-B. Huang, and J.-W. Shi, “Photonic High-Power Continuous Wave THz-Wave Generation by Using Flip-Chip Packaged Uni-Traveling Carrier Photodiodes and a Femtosecond Optical Pulse Generator,” J. Lightwave Technol. 34(4), 1387–1397 (2016).
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C.-C. Chen, I.-C. Hsieh, S.-D. Yang, and C.-B. Huang, “Polarization line-by-line pulse shaping for the implementation of vectorial temporal Talbot effect,” Opt. Express 20(24), 27062–27070 (2012).
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F.-M. Kuo, C.-B. Huang, J.-W. Shi, N.-W. Chen, H.-P. Chuang, J. E. Bowers, and C.-L. Pan, “Remotely up-converted 20-Gbit/s error-free wireless on-off-keying data transmission at W-band using an ultra-wideband photonic transmitter-mixer,” IEEE Photonics J. 3(2), 209–219 (2011).
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J. Caraquitena, Z. Jiang, D. E. Leaird, and A. M. Weiner, “Tunable pulse repetition-rate multiplication using phase-only line-by-line pulse shaping,” Opt. Lett. 32(6), 716–718 (2007).
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F.-M. Kuo, C.-B. Huang, J.-W. Shi, N.-W. Chen, H.-P. Chuang, J. E. Bowers, and C.-L. Pan, “Remotely up-converted 20-Gbit/s error-free wireless on-off-keying data transmission at W-band using an ultra-wideband photonic transmitter-mixer,” IEEE Photonics J. 3(2), 209–219 (2011).
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V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6(3), 186–194 (2012).
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J. Caraquitena, Z. Jiang, D. E. Leaird, and A. M. Weiner, “Tunable pulse repetition-rate multiplication using phase-only line-by-line pulse shaping,” Opt. Lett. 32(6), 716–718 (2007).
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K. Wörhoff, R. G. Heideman, A. Leinse, and M. Hoekman, “TriPleX: a versatile dielectric photonic platform,” Adv. Opt. Technol. 4(2), 189–207 (2015).

Leuthold, J.

D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010).
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Y. Li, J. Li, H. Yu, H. Yu, H. Chen, S. Yang, and M. Chen, “On-chip photonic microsystem for optical signal processing based on silicon and silicon nitride platforms,” Adv. Opt. Technol. 7(1-2), 81–101 (2018).
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D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010).
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R. Maram, J. Van Howe, M. Li, and J. Azaña, “Noiseless intensity amplification of repetitive signals by coherent addition using the temporal Talbot effect,” Nat. Commun. 5(1), 5163 (2014).
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Y. Li, J. Li, H. Yu, H. Yu, H. Chen, S. Yang, and M. Chen, “On-chip photonic microsystem for optical signal processing based on silicon and silicon nitride platforms,” Adv. Opt. Technol. 7(1-2), 81–101 (2018).
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S. Liao, Y. Ding, J. Dong, T. Yang, X. Chen, D. Gao, and X. Zhang, “Arbitrary waveform generator and differentiator employing an integrated optical pulse shaper,” Opt. Express 23(9), 12161–12173 (2015).
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Liu, H.-Y.

J.-M. Wun, H.-Y. Liu, Y.-L. Zeng, S.-D. Yang, C.-L. Pan, C.-B. Huang, and J.-W. Shi, “Photonic High-Power Continuous Wave THz-Wave Generation by Using Flip-Chip Packaged Uni-Traveling Carrier Photodiodes and a Femtosecond Optical Pulse Generator,” J. Lightwave Technol. 34(4), 1387–1397 (2016).
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V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6(3), 186–194 (2012).
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Lowery, A. J.

Y. Xie, L. Zhuang, and A. J. Lowery, “Picosecond optical pulse processing using a terahertz-bandwidth reconfigurable photonic integrated circuit,” Nanophotonics 7(5), 837–852 (2018).
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Z. Geng, Y. Xie, L. Zhuang, M. Burla, M. Hoekman, C. G. H. Roeloffzen, and A. J. Lowery, “Photonic integrated circuit implementation of a sub-GHz-selectivity frequency comb filter for optical clock multiplication,” Opt. Express 25(22), 27635–27645 (2017).
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A. Malacarne and J. Azaña, “Discretely tunable comb spacing of a frequency comb by multilevel phase modulation of a periodic pulse train,” Opt. Express 21(4), 4139–4144 (2013).
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L. R. Cortés, M. Seghilani, R. Maram, and J. Azaña, “Full-field broadband invisibility through reversible wave frequency-spectrum control,” Optica 5(7), 779–786 (2018).
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M. Seghilani, R. Maram, and J. Azaña, “Mitigating nonlinear propagation impairments of ultrashort pulses by fractional temporal self-imaging,” Opt. Lett. 42(4), 879–882 (2017).
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C. R. Fernández-Pousa, R. Maram, and J. Azaña, “CW-to-pulse conversion using temporal Talbot array illuminators,” Opt. Lett. 42(13), 2427–2430 (2017).
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R. Maram, L. R. Cortés, and J. Azaña, “Programmable fiber-optics pulse repetition-rate multiplier,” J. Lightwave Technol. 34(2), 448–455 (2016).
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R. Maram, J. Van Howe, M. Li, and J. Azaña, “Noiseless intensity amplification of repetitive signals by coherent addition using the temporal Talbot effect,” Nat. Commun. 5(1), 5163 (2014).
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D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010).
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S. Min, Y. Zhao, and S. Fleming, “Repetition rate multiplication in figure-eight fibre laser with 3 dB couplers,” Opt. Commun. 277(2), 411–413 (2007).
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J. Caraquitena, M. Beltrán, R. Llorente, J. Martí, and M. A. Muriel, “Spectral self-imaging effect by time-domain multilevel phase modulation of a periodic pulse train,” Opt. Lett. 36(6), 858–860 (2011).
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Narkiss, N.

D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010).
[Crossref] [PubMed]

Pan, C.-L.

J.-M. Wun, H.-Y. Liu, Y.-L. Zeng, S.-D. Yang, C.-L. Pan, C.-B. Huang, and J.-W. Shi, “Photonic High-Power Continuous Wave THz-Wave Generation by Using Flip-Chip Packaged Uni-Traveling Carrier Photodiodes and a Femtosecond Optical Pulse Generator,” J. Lightwave Technol. 34(4), 1387–1397 (2016).
[Crossref]

F.-M. Kuo, C.-B. Huang, J.-W. Shi, N.-W. Chen, H.-P. Chuang, J. E. Bowers, and C.-L. Pan, “Remotely up-converted 20-Gbit/s error-free wireless on-off-keying data transmission at W-band using an ultra-wideband photonic transmitter-mixer,” IEEE Photonics J. 3(2), 209–219 (2011).
[Crossref]

Preciado, M. A.

M. A. Preciado and M. A. Muriel, “All-pass optical structures for repetition rate multiplication,” Opt. Express 16(15), 11162–11168 (2008).
[Crossref] [PubMed]

Roeloffzen, C. G. H.

Z. Geng, Y. Xie, L. Zhuang, M. Burla, M. Hoekman, C. G. H. Roeloffzen, and A. J. Lowery, “Photonic integrated circuit implementation of a sub-GHz-selectivity frequency comb filter for optical clock multiplication,” Opt. Express 25(22), 27635–27645 (2017).
[Crossref] [PubMed]

Sahara, A.

M. Nakazawa, E. Yoshida, T. Yamamoto, E. Yamada, and A. Sahara, “TDM single channel 640Gbit/s transmission experiment over 60 km using 400fs pulse train and walk-off free, dispersion flattened nonlinear optical loop mirror,” Electron. Lett. 34(9), 907–908 (1998).
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Schneider, T.

M. A. Soto, M. Alem, M. Amin Shoaie, A. Vedadi, C. S. Brès, L. Thévenaz, and T. Schneider, “Optical sinc-shaped Nyquist pulses of exceptional quality,” Nat. Commun. 4(1), 2898 (2013).
[Crossref] [PubMed]

Seghilani, M.

L. R. Cortés, M. Seghilani, R. Maram, and J. Azaña, “Full-field broadband invisibility through reversible wave frequency-spectrum control,” Optica 5(7), 779–786 (2018).
[Crossref]

M. Seghilani, R. Maram, and J. Azaña, “Mitigating nonlinear propagation impairments of ultrashort pulses by fractional temporal self-imaging,” Opt. Lett. 42(4), 879–882 (2017).
[Crossref] [PubMed]

Shi, J.-W.

J.-M. Wun, H.-Y. Liu, Y.-L. Zeng, S.-D. Yang, C.-L. Pan, C.-B. Huang, and J.-W. Shi, “Photonic High-Power Continuous Wave THz-Wave Generation by Using Flip-Chip Packaged Uni-Traveling Carrier Photodiodes and a Femtosecond Optical Pulse Generator,” J. Lightwave Technol. 34(4), 1387–1397 (2016).
[Crossref]

F.-M. Kuo, C.-B. Huang, J.-W. Shi, N.-W. Chen, H.-P. Chuang, J. E. Bowers, and C.-L. Pan, “Remotely up-converted 20-Gbit/s error-free wireless on-off-keying data transmission at W-band using an ultra-wideband photonic transmitter-mixer,” IEEE Photonics J. 3(2), 209–219 (2011).
[Crossref]

Sigurdsson, G.

D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010).
[Crossref] [PubMed]

Soto, M. A.

M. A. Soto, M. Alem, M. Amin Shoaie, A. Vedadi, C. S. Brès, L. Thévenaz, and T. Schneider, “Optical sinc-shaped Nyquist pulses of exceptional quality,” Nat. Commun. 4(1), 2898 (2013).
[Crossref] [PubMed]

Supradeepa, V. R.

V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6(3), 186–194 (2012).
[Crossref]

Teschke, M.

D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010).
[Crossref] [PubMed]

Thévenaz, L.

M. A. Soto, M. Alem, M. Amin Shoaie, A. Vedadi, C. S. Brès, L. Thévenaz, and T. Schneider, “Optical sinc-shaped Nyquist pulses of exceptional quality,” Nat. Commun. 4(1), 2898 (2013).
[Crossref] [PubMed]

Torres-Company, V.

V. Torres-Company and A. M. Weiner, “Optical frequency comb technology for ultra-broadband radiofrequency photonics,” Laser Photonics Rev. 8(3), 368–393 (2014).
[Crossref]

V. Torres-Company, A. J. Metcalf, D. Leaird, and A. M. Weiner, “Multichannel radio-frequency arbitrary waveform generation based on multiwavelength comb switching and 2-D line-by-line pulse shaping,” IEEE Photonics Technol. Lett. 24(11), 891–893 (2012).
[Crossref]

Van Howe, J.

R. Maram, J. Van Howe, M. Li, and J. Azaña, “Noiseless intensity amplification of repetitive signals by coherent addition using the temporal Talbot effect,” Nat. Commun. 5(1), 5163 (2014).
[Crossref] [PubMed]

Vedadi, A.

M. A. Soto, M. Alem, M. Amin Shoaie, A. Vedadi, C. S. Brès, L. Thévenaz, and T. Schneider, “Optical sinc-shaped Nyquist pulses of exceptional quality,” Nat. Commun. 4(1), 2898 (2013).
[Crossref] [PubMed]

Weiner, A. M.

V. Torres-Company and A. M. Weiner, “Optical frequency comb technology for ultra-broadband radiofrequency photonics,” Laser Photonics Rev. 8(3), 368–393 (2014).
[Crossref]

V. Torres-Company, A. J. Metcalf, D. Leaird, and A. M. Weiner, “Multichannel radio-frequency arbitrary waveform generation based on multiwavelength comb switching and 2-D line-by-line pulse shaping,” IEEE Photonics Technol. Lett. 24(11), 891–893 (2012).
[Crossref]

V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6(3), 186–194 (2012).
[Crossref]

J. Caraquitena, Z. Jiang, D. E. Leaird, and A. M. Weiner, “Tunable pulse repetition-rate multiplication using phase-only line-by-line pulse shaping,” Opt. Lett. 32(6), 716–718 (2007).
[Crossref] [PubMed]

Winter, M.

D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010).
[Crossref] [PubMed]

Wörhoff, K.

K. Wörhoff, R. G. Heideman, A. Leinse, and M. Hoekman, “TriPleX: a versatile dielectric photonic platform,” Adv. Opt. Technol. 4(2), 189–207 (2015).

Worms, K.

D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010).
[Crossref] [PubMed]

Wu, R.

V. R. Supradeepa, C. M. Long, R. Wu, F. Ferdous, E. Hamidi, D. E. Leaird, and A. M. Weiner, “Comb-based radiofrequency photonic filters with rapid tunability and high selectivity,” Nat. Photonics 6(3), 186–194 (2012).
[Crossref]

Wun, J.-M.

J.-M. Wun, H.-Y. Liu, Y.-L. Zeng, S.-D. Yang, C.-L. Pan, C.-B. Huang, and J.-W. Shi, “Photonic High-Power Continuous Wave THz-Wave Generation by Using Flip-Chip Packaged Uni-Traveling Carrier Photodiodes and a Femtosecond Optical Pulse Generator,” J. Lightwave Technol. 34(4), 1387–1397 (2016).
[Crossref]

Xie, Y.

Y. Xie, L. Zhuang, and A. J. Lowery, “Picosecond optical pulse processing using a terahertz-bandwidth reconfigurable photonic integrated circuit,” Nanophotonics 7(5), 837–852 (2018).
[Crossref]

Z. Geng, Y. Xie, L. Zhuang, M. Burla, M. Hoekman, C. G. H. Roeloffzen, and A. J. Lowery, “Photonic integrated circuit implementation of a sub-GHz-selectivity frequency comb filter for optical clock multiplication,” Opt. Express 25(22), 27635–27645 (2017).
[Crossref] [PubMed]

Yamada, E.

M. Nakazawa, E. Yoshida, T. Yamamoto, E. Yamada, and A. Sahara, “TDM single channel 640Gbit/s transmission experiment over 60 km using 400fs pulse train and walk-off free, dispersion flattened nonlinear optical loop mirror,” Electron. Lett. 34(9), 907–908 (1998).
[Crossref]

Yamamoto, T.

M. Nakazawa, E. Yoshida, T. Yamamoto, E. Yamada, and A. Sahara, “TDM single channel 640Gbit/s transmission experiment over 60 km using 400fs pulse train and walk-off free, dispersion flattened nonlinear optical loop mirror,” Electron. Lett. 34(9), 907–908 (1998).
[Crossref]

Yang, S.

Y. Li, J. Li, H. Yu, H. Yu, H. Chen, S. Yang, and M. Chen, “On-chip photonic microsystem for optical signal processing based on silicon and silicon nitride platforms,” Adv. Opt. Technol. 7(1-2), 81–101 (2018).
[Crossref]

Yang, S.-D.

J.-M. Wun, H.-Y. Liu, Y.-L. Zeng, S.-D. Yang, C.-L. Pan, C.-B. Huang, and J.-W. Shi, “Photonic High-Power Continuous Wave THz-Wave Generation by Using Flip-Chip Packaged Uni-Traveling Carrier Photodiodes and a Femtosecond Optical Pulse Generator,” J. Lightwave Technol. 34(4), 1387–1397 (2016).
[Crossref]

C.-C. Chen, I.-C. Hsieh, S.-D. Yang, and C.-B. Huang, “Polarization line-by-line pulse shaping for the implementation of vectorial temporal Talbot effect,” Opt. Express 20(24), 27062–27070 (2012).
[Crossref] [PubMed]

Yang, T.

S. Liao, Y. Ding, J. Dong, T. Yang, X. Chen, D. Gao, and X. Zhang, “Arbitrary waveform generator and differentiator employing an integrated optical pulse shaper,” Opt. Express 23(9), 12161–12173 (2015).
[Crossref] [PubMed]

Yao, J.

Y. Dai and J. Yao, “Nonuniformly-spaced photonic microwave delayline filter,” Opt. Express 16(7), 4713–4718 (2008).
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Figures (6)

Fig. 1
Fig. 1 Schematic diagram of the optical tapped delay line structure. T0 is the unit delay time, and ϕn (n = 0, 1, 2,…, N-1) are the phase applied to phase tuning elements.
Fig. 2
Fig. 2 (a) Power (top) and phase (bottom) response of the spectral amplitude filtering, when the phase shifters are configured all in-phase, [0,0,0,0,0,0,0,0]. (b) Optical spectra (top) and temporal waveforms (bottom) at the input and output the filter.
Fig. 3
Fig. 3 (a) Power (top) and phase (bottom) response of the spectral phase-only filtering, when the phase shifters are configured by Talbot phase of N = 8, [0,π/8,π/2,9π/8,0,9π/8,π/2,π/8]. (b) Optical spectra (top) and temporal waveforms (bottom) at the input and output the filter.
Fig. 4
Fig. 4 Left: power (top) and phase (bottom) response of the spectral phase and amplitude filtering; Right: optical spectra (top) and temporal waveforms (bottom) at the output the filter. The phase shifters are configured by repeated Talbot phase of (a) N = 4, [0,π/4,π,π/4,0,π/4,π,π/4] and (b) N = 2, [0,π/2,0,π/2, 0,π/2,0,π/2].
Fig. 5
Fig. 5 (a) Experimental setup. MZM: Mach-Zehnder modulator; φ: electrical phase shifter; DLI: delay line interferometer; OSA: optical spectrum analyzer; OSO: optical sampling oscilloscope. (b) Optical spectra (top) and temporal waveforms (bottom) measured at the input and two output ports of the DLI.
Fig. 6
Fig. 6 Left to right: Power and phase response of the proposed filter, output optical spectra and temporal waveforms in the presence of (a) power imbalance (b) delay line length inaccuracy, and (c) phase error. The pink markers and traces correspond to 200 superimposed variations with predefined standard deviation.

Equations (15)

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H(f)= 1 N n=0 N1 exp( i2πnf T 0 i ϕ n )
ϕ n ={ π n 2 N if N0 (mod 2) 2π n 2 N if N1 (mod 2)
H( f k = k N T 0 )= 1 N n=0 N1 exp( i 2πnk N i ϕ n ) = 1 N exp( iθ+is ϕ k ) ={ 1 N exp( i π 4 +i ϕ k ) if N0 (mod 2) 1 N exp( i 3N+1 4 ϕ k ) if N1 (mod 4) 1 N exp( i π 2 +i N+1 4 ϕ k ) if N3 (mod 4)
E in ( f )= E 0 K k=0 K1 A( k )δ( fk f m )
E out (f)= E in (f)H(f)= E 0 KN k=0 K1 A( k )exp( iθ+is ϕ k )δ( fk f m )
n=0 N1 exp( iπ n 2 l+nm N ) = N l exp( iπ m 2 Nl 4Nl ) n=0 l1 exp( iπ n 2 N+nm l )
n=0 N1 exp ( i π n 2 N )= N exp( i π 4 )
n=0 N1 exp( i 2πnk+π n 2 N ) =exp( i π k 2 N ) n=0 N1 exp( i π (n+k) 2 N ) = N exp( i π 4 +i π k 2 N )
n=0 N1 exp( iπ 2 n 2 +2nk N ) = N 2 exp( i π k 2 2N i π 4 )(1+exp( iπ N+2k 2 ))
1+exp( iπ N+2k 2 )={ 2 exp( i π 4 ) if k0 (mod 2) 2 exp( i π 4 ) if k1 (mod 2)
exp( iπ k 2 2N )={ exp( iπ k 2 2N +i 3π 2 k 2 ) if k0 (mod 2) exp( iπ k 2 2N +i 3π 2 ( k 2 1) ) if k1 (mod 2)
n=0 N1 exp( iπ 2 n 2 +2nk N ) = N exp( iπ k 2 2N +i 3π 2 k 2 ) = N exp( i 3N+1 4 2π k 2 N )
1+exp( iπ N+2k 2 )={ 2 exp( i π 4 ) if k0 (mod 2) 2 exp( i π 4 ) if k1 (mod 2)
exp( iπ k 2 2N )={ exp( iπ k 2 2N +i π 2 k 2 ) if k0 (mod 2) exp( iπ k 2 2N +i π 2 ( k 2 1) ) if k1 (mod 2)
n=0 N1 exp( iπ 2 n 2 +2nk N ) = N exp( i π 2 +iπ k 2 2N +i π 2 k 2 ) = N exp( i π 2 +i N+1 4 2π k 2 N )

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