Abstract

In this paper, the mathematical description of the temporal self-imaging effect is studied, focusing on the situation in which the train of pulses to be dispersed has been previously periodically modulated in phase and amplitude. It is demonstrated that, for each input pulse and for some specific values of the chromatic dispersion, a subtrain of optical pulses is generated whose envelope is determined by the Discrete Fourier Transform of the modulating coefficients. The mathematical results are confirmed by simulations of various examples and some limits on the realization of the theory are commented.

© 2014 Optical Society of America

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  1. L. K. Oxenlowe, H. Ji, M. Galili, M. H. Pu, H. Hu, H. C. H. Mulvad, K. Yvind, J. M. Hvam, A. T. Clausen, and P. Jeppesen, “Silicon photonics for signal processing of Tbit/s serial data signals,” IEEE J. Sel. Top. Quantum Electron. 18(2), 996–1005 (2012).
    [Crossref]
  2. S. J. B. Yoo, R. P. Scott, D. J. Geisler, N. K. Fontaine, and F. M. Soares, “Terahertz information and signal processing by RF-photonics,” IEEE Tran. Terahertz Sci. Technol. 2(2), 167–176 (2012).
    [Crossref]
  3. P. J. Delfyett, I. Ozdur, N. Hoghooghi, M. Akbulut, J. Davila-Rodriguez, and S. Bhooplapur, “Advanced ultrafast technologies based on optical frequency combs,” IEEE J. Sel. Top. Quantum Electron. 18(1), 258–274 (2012).
    [Crossref]
  4. A. E. Willner, O. F. Yilmaz, J. A. Wang, X. X. Wu, A. Bogoni, L. Zhang, and S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17(2), 320–332 (2011).
    [Crossref]
  5. R. S. Tucker and K. Hinton, “Energy consumption and energy density in optical and electronic signal processing,” IEEE Photonics J. 3(5), 821–833 (2011).
    [Crossref]
  6. J. W. Goodman, Introduction to Fourier Optics (Roberts and Co., 2005).
  7. A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19(3), 161–237 (1995).
    [Crossref]
  8. A. M. Weiner and A. M. Kan’an, “Femtosecond pulse shaping for synthesis, processing, and time-to-space conversion of ultrafast optical waveforms,” IEEE J. Sel. Top. Quantum Electron. 4(2), 317–331 (1998).
    [Crossref]
  9. B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30(8), 1951–1963 (1994).
    [Crossref]
  10. A. Papoulis, Systems and Transforms With Applications in Optics (McGraw-Hill, 1968).
  11. S. A. Akhmanov, A. P. Sukhoruk, and A. S. Chirkin, “Nonstationary phenomena and space-time analogy in nonlinear optics,” Sov. Phys. JETP-USSR 28, 748 (1969).
  12. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5(9), 454–458 (1969).
    [Crossref]
  13. B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14(12), 630–632 (1989).
    [Crossref] [PubMed]
  14. C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging - Part I: System configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).
  15. C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging - Part II: System performance,” IEEE J. Quantum Electron. 36, 649–655 (2000).
  16. 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(10), 1047–1049 (2008).
    [Crossref] [PubMed]
  17. J. van Howe and C. Xu, “Ultrafast optical signal processing based upon space-time dualities,” J. Lightwave Technol. 24(7), 2649–2662 (2006).
    [Crossref]
  18. M. A. Foster, R. Salem, and A. L. Gaeta, “Ultrahigh-speed optical processing using space-time duality,” Opt. Photon. News 22(5), 29–35 (2011).
    [Crossref]
  19. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
    [Crossref] [PubMed]
  20. A. V. Mamaev and M. Saffman, “Selection of unstable patterns and control of optical turbulence by Fourier plane filtering,” Phys. Rev. Lett. 80(16), 3499–3502 (1998).
    [Crossref]
  21. M. A. Muriel, J. Azaña, and A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. 24(1), 1–3 (1999).
    [Crossref] [PubMed]
  22. J. Azana and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).
  23. R. Salem, M. A. Foster, A. C. Turner-Foster, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “High-speed optical sampling using a silicon-chip temporal magnifier,” Opt. Express 17(6), 4324–4329 (2009).
    [Crossref] [PubMed]
  24. K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7(2), 102–112 (2013).
    [Crossref]
  25. 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(3), 882–884 (2004).
    [Crossref]
  26. R. Salem, M. A. Foster, and A. L. Gaeta, “Application of space–time duality to ultrahigh-speed optical signal processing,” Adv. Opt. Photon. 5(3), 274–317 (2013).
    [Crossref]
  27. S. Thomas, A. Malacarne, F. Fresi, L. Poti, and J. Azana, “Fiber-based programmable picosecond optical pulse shaper,” J. Lightwave Technol. 28(12), 1832–1843 (2010).
    [Crossref]
  28. J. Azaña, “Design specifications of time-domain spectral shaping optical system based on dispersion and temporal modulation,” Electron. Lett. 39(21), 1530–1532 (2003).
    [Crossref]
  29. C. Wang and J. P. Yao, “Chirped microwave pulse generation based on optical spectral shaping and wavelength-to-time mapping using a Sagnac loop mirror incorporating a chirped fiber Bragg grating,” J. Lightwave Technol. 27(16), 3336–3341 (2009).
    [Crossref]
  30. H. Chi and J. Yao, “All-fiber chirped microwave pulses generation based on spectral shaping and wavelength-to-time conversion,” IEEE Trans. Microwave Theory 55(9), 1958–1963 (2007).
    [Crossref]
  31. Y. Park, T. J. Ahn, J. C. Kieffer, and J. Azaña, “Optical frequency domain reflectometry based on real-time Fourier transformation,” Opt. Express 15(8), 4597–4616 (2007).
    [Crossref] [PubMed]
  32. H. F. Talbot, “Facts relating to optical science no. IV,” Philos. Mag. 9, 401–407 (1836).
  33. L. Rayleigh, “On copying diffraction gratings and on some phenomenon connected therewith,” Philos. Mag. 11(67), 196–205 (1881).
    [Crossref]
  34. M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
    [Crossref]
  35. M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
    [Crossref] [PubMed]
  36. J. Wen, Y. Zhang, and M. Xiao, “The Talbot effect: recent advances in classical optics, nonlinear optics, and quantum optics,” Adv. Opt. Photon. 5(1), 83–130 (2013).
    [Crossref]
  37. T. Jannson and J. Jannson, “Temporal self-imaging effect in single-mode fibers,” J. Opt. Soc. Am. 71, 1373–1376 (1981).
  38. J. Azaña and M. A. Muriel, “Temporal Talbot effect in fiber gratings and its applications,” Appl. Opt. 38(32), 6700–6704 (1999).
    [Crossref] [PubMed]
  39. 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]
  40. J. Azaña and L. R. Chen, “General temporal self-imaging phenomena,” J. Opt. Soc. Am. B 20(7), 1447–1458 (2003).
    [Crossref]
  41. D. Pudo and L. R. Chen, “Tunable passive all-optical pulse repetition rate multiplier using fiber Bragg gratings,” J. Lightwave Technol. 23(4), 1729–1733 (2005).
    [Crossref]
  42. J. Caraquitena and J. Martí, “High-rate pulse-train generation by phase-only filtering of an electrooptic frequency comb: Analysis and optimization,” Opt. Commun. 282(18), 3686–3692 (2009).
    [Crossref]
  43. 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]

2013 (3)

2012 (3)

L. K. Oxenlowe, H. Ji, M. Galili, M. H. Pu, H. Hu, H. C. H. Mulvad, K. Yvind, J. M. Hvam, A. T. Clausen, and P. Jeppesen, “Silicon photonics for signal processing of Tbit/s serial data signals,” IEEE J. Sel. Top. Quantum Electron. 18(2), 996–1005 (2012).
[Crossref]

S. J. B. Yoo, R. P. Scott, D. J. Geisler, N. K. Fontaine, and F. M. Soares, “Terahertz information and signal processing by RF-photonics,” IEEE Tran. Terahertz Sci. Technol. 2(2), 167–176 (2012).
[Crossref]

P. J. Delfyett, I. Ozdur, N. Hoghooghi, M. Akbulut, J. Davila-Rodriguez, and S. Bhooplapur, “Advanced ultrafast technologies based on optical frequency combs,” IEEE J. Sel. Top. Quantum Electron. 18(1), 258–274 (2012).
[Crossref]

2011 (3)

A. E. Willner, O. F. Yilmaz, J. A. Wang, X. X. Wu, A. Bogoni, L. Zhang, and S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17(2), 320–332 (2011).
[Crossref]

R. S. Tucker and K. Hinton, “Energy consumption and energy density in optical and electronic signal processing,” IEEE Photonics J. 3(5), 821–833 (2011).
[Crossref]

M. A. Foster, R. Salem, and A. L. Gaeta, “Ultrahigh-speed optical processing using space-time duality,” Opt. Photon. News 22(5), 29–35 (2011).
[Crossref]

2010 (1)

2009 (3)

2008 (1)

2007 (3)

2006 (1)

2005 (1)

2004 (1)

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(3), 882–884 (2004).
[Crossref]

2003 (2)

J. Azaña, “Design specifications of time-domain spectral shaping optical system based on dispersion and temporal modulation,” Electron. Lett. 39(21), 1530–1532 (2003).
[Crossref]

J. Azaña and L. R. Chen, “General temporal self-imaging phenomena,” J. Opt. Soc. Am. B 20(7), 1447–1458 (2003).
[Crossref]

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 (3)

J. Azana and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging - Part I: System configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging - Part II: System performance,” IEEE J. Quantum Electron. 36, 649–655 (2000).

1999 (2)

1998 (2)

A. V. Mamaev and M. Saffman, “Selection of unstable patterns and control of optical turbulence by Fourier plane filtering,” Phys. Rev. Lett. 80(16), 3499–3502 (1998).
[Crossref]

A. M. Weiner and A. M. Kan’an, “Femtosecond pulse shaping for synthesis, processing, and time-to-space conversion of ultrafast optical waveforms,” IEEE J. Sel. Top. Quantum Electron. 4(2), 317–331 (1998).
[Crossref]

1996 (1)

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
[Crossref]

1995 (3)

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[Crossref] [PubMed]

P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
[Crossref] [PubMed]

A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19(3), 161–237 (1995).
[Crossref]

1994 (1)

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30(8), 1951–1963 (1994).
[Crossref]

1989 (1)

1981 (1)

1969 (2)

S. A. Akhmanov, A. P. Sukhoruk, and A. S. Chirkin, “Nonstationary phenomena and space-time analogy in nonlinear optics,” Sov. Phys. JETP-USSR 28, 748 (1969).

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5(9), 454–458 (1969).
[Crossref]

1881 (1)

L. Rayleigh, “On copying diffraction gratings and on some phenomenon connected therewith,” Philos. Mag. 11(67), 196–205 (1881).
[Crossref]

1836 (1)

H. F. Talbot, “Facts relating to optical science no. IV,” Philos. Mag. 9, 401–407 (1836).

Ahn, T. J.

Akbulut, M.

P. J. Delfyett, I. Ozdur, N. Hoghooghi, M. Akbulut, J. Davila-Rodriguez, and S. Bhooplapur, “Advanced ultrafast technologies based on optical frequency combs,” IEEE J. Sel. Top. Quantum Electron. 18(1), 258–274 (2012).
[Crossref]

Akhmanov, S. A.

S. A. Akhmanov, A. P. Sukhoruk, and A. S. Chirkin, “Nonstationary phenomena and space-time analogy in nonlinear optics,” Sov. Phys. JETP-USSR 28, 748 (1969).

Azana, J.

S. Thomas, A. Malacarne, F. Fresi, L. Poti, and J. Azana, “Fiber-based programmable picosecond optical pulse shaper,” J. Lightwave Technol. 28(12), 1832–1843 (2010).
[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(3), 882–884 (2004).
[Crossref]

J. Azana and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).

Azaña, J.

Bennett, C. V.

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging - Part I: System configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging - Part II: System performance,” IEEE J. Quantum Electron. 36, 649–655 (2000).

Berger, N. K.

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(3), 882–884 (2004).
[Crossref]

Berry, M. V.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
[Crossref]

Bhooplapur, S.

P. J. Delfyett, I. Ozdur, N. Hoghooghi, M. Akbulut, J. Davila-Rodriguez, and S. Bhooplapur, “Advanced ultrafast technologies based on optical frequency combs,” IEEE J. Sel. Top. Quantum Electron. 18(1), 258–274 (2012).
[Crossref]

Bogoni, A.

A. E. Willner, O. F. Yilmaz, J. A. Wang, X. X. Wu, A. Bogoni, L. Zhang, and S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17(2), 320–332 (2011).
[Crossref]

Caraquitena, J.

J. Caraquitena and J. Martí, “High-rate pulse-train generation by phase-only filtering of an electrooptic frequency comb: Analysis and optimization,” Opt. Commun. 282(18), 3686–3692 (2009).
[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]

Carballar, A.

Chapman, M. S.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[Crossref] [PubMed]

Chen, L. R.

Chi, H.

H. Chi and J. Yao, “All-fiber chirped microwave pulses generation based on spectral shaping and wavelength-to-time conversion,” IEEE Trans. Microwave Theory 55(9), 1958–1963 (2007).
[Crossref]

Chirkin, A. S.

S. A. Akhmanov, A. P. Sukhoruk, and A. S. Chirkin, “Nonstationary phenomena and space-time analogy in nonlinear optics,” Sov. Phys. JETP-USSR 28, 748 (1969).

Clausen, A. T.

L. K. Oxenlowe, H. Ji, M. Galili, M. H. Pu, H. Hu, H. C. H. Mulvad, K. Yvind, J. M. Hvam, A. T. Clausen, and P. Jeppesen, “Silicon photonics for signal processing of Tbit/s serial data signals,” IEEE J. Sel. Top. Quantum Electron. 18(2), 996–1005 (2012).
[Crossref]

Davila-Rodriguez, J.

P. J. Delfyett, I. Ozdur, N. Hoghooghi, M. Akbulut, J. Davila-Rodriguez, and S. Bhooplapur, “Advanced ultrafast technologies based on optical frequency combs,” IEEE J. Sel. Top. Quantum Electron. 18(1), 258–274 (2012).
[Crossref]

Delfyett, P. J.

P. J. Delfyett, I. Ozdur, N. Hoghooghi, M. Akbulut, J. Davila-Rodriguez, and S. Bhooplapur, “Advanced ultrafast technologies based on optical frequency combs,” IEEE J. Sel. Top. Quantum Electron. 18(1), 258–274 (2012).
[Crossref]

Ekstrom, C. R.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[Crossref] [PubMed]

Fischer, B.

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(3), 882–884 (2004).
[Crossref]

Fontaine, N. K.

S. J. B. Yoo, R. P. Scott, D. J. Geisler, N. K. Fontaine, and F. M. Soares, “Terahertz information and signal processing by RF-photonics,” IEEE Tran. Terahertz Sci. Technol. 2(2), 167–176 (2012).
[Crossref]

Foster, M. A.

Fresi, F.

Gaeta, A. L.

Galili, M.

L. K. Oxenlowe, H. Ji, M. Galili, M. H. Pu, H. Hu, H. C. H. Mulvad, K. Yvind, J. M. Hvam, A. T. Clausen, and P. Jeppesen, “Silicon photonics for signal processing of Tbit/s serial data signals,” IEEE J. Sel. Top. Quantum Electron. 18(2), 996–1005 (2012).
[Crossref]

Geisler, D. J.

S. J. B. Yoo, R. P. Scott, D. J. Geisler, N. K. Fontaine, and F. M. Soares, “Terahertz information and signal processing by RF-photonics,” IEEE Tran. Terahertz Sci. Technol. 2(2), 167–176 (2012).
[Crossref]

Geraghty, D. F.

Goda, K.

K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7(2), 102–112 (2013).
[Crossref]

Hammond, T. D.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[Crossref] [PubMed]

Hinton, K.

R. S. Tucker and K. Hinton, “Energy consumption and energy density in optical and electronic signal processing,” IEEE Photonics J. 3(5), 821–833 (2011).
[Crossref]

Hoghooghi, N.

P. J. Delfyett, I. Ozdur, N. Hoghooghi, M. Akbulut, J. Davila-Rodriguez, and S. Bhooplapur, “Advanced ultrafast technologies based on optical frequency combs,” IEEE J. Sel. Top. Quantum Electron. 18(1), 258–274 (2012).
[Crossref]

Hu, H.

L. K. Oxenlowe, H. Ji, M. Galili, M. H. Pu, H. Hu, H. C. H. Mulvad, K. Yvind, J. M. Hvam, A. T. Clausen, and P. Jeppesen, “Silicon photonics for signal processing of Tbit/s serial data signals,” IEEE J. Sel. Top. Quantum Electron. 18(2), 996–1005 (2012).
[Crossref]

Hvam, J. M.

L. K. Oxenlowe, H. Ji, M. Galili, M. H. Pu, H. Hu, H. C. H. Mulvad, K. Yvind, J. M. Hvam, A. T. Clausen, and P. Jeppesen, “Silicon photonics for signal processing of Tbit/s serial data signals,” IEEE J. Sel. Top. Quantum Electron. 18(2), 996–1005 (2012).
[Crossref]

Jalali, B.

K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7(2), 102–112 (2013).
[Crossref]

Jannson, J.

Jannson, T.

Javidi, B.

Jeppesen, P.

L. K. Oxenlowe, H. Ji, M. Galili, M. H. Pu, H. Hu, H. C. H. Mulvad, K. Yvind, J. M. Hvam, A. T. Clausen, and P. Jeppesen, “Silicon photonics for signal processing of Tbit/s serial data signals,” IEEE J. Sel. Top. Quantum Electron. 18(2), 996–1005 (2012).
[Crossref]

Ji, H.

L. K. Oxenlowe, H. Ji, M. Galili, M. H. Pu, H. Hu, H. C. H. Mulvad, K. Yvind, J. M. Hvam, A. T. Clausen, and P. Jeppesen, “Silicon photonics for signal processing of Tbit/s serial data signals,” IEEE J. Sel. Top. Quantum Electron. 18(2), 996–1005 (2012).
[Crossref]

Jiang, Z.

Kan’an, A. M.

A. M. Weiner and A. M. Kan’an, “Femtosecond pulse shaping for synthesis, processing, and time-to-space conversion of ultrafast optical waveforms,” IEEE J. Sel. Top. Quantum Electron. 4(2), 317–331 (1998).
[Crossref]

Kieffer, J. C.

Klein, S.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
[Crossref]

Kolner, B. H.

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging - Part II: System performance,” IEEE J. Quantum Electron. 36, 649–655 (2000).

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging - Part I: System configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30(8), 1951–1963 (1994).
[Crossref]

B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14(12), 630–632 (1989).
[Crossref] [PubMed]

Leaird, D. E.

Levit, B.

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(3), 882–884 (2004).
[Crossref]

Lipson, M.

Malacarne, A.

Mamaev, A. V.

A. V. Mamaev and M. Saffman, “Selection of unstable patterns and control of optical turbulence by Fourier plane filtering,” Phys. Rev. Lett. 80(16), 3499–3502 (1998).
[Crossref]

Martí, J.

J. Caraquitena and J. Martí, “High-rate pulse-train generation by phase-only filtering of an electrooptic frequency comb: Analysis and optimization,” Opt. Commun. 282(18), 3686–3692 (2009).
[Crossref]

Mulvad, H. C. H.

L. K. Oxenlowe, H. Ji, M. Galili, M. H. Pu, H. Hu, H. C. H. Mulvad, K. Yvind, J. M. Hvam, A. T. Clausen, and P. Jeppesen, “Silicon photonics for signal processing of Tbit/s serial data signals,” IEEE J. Sel. Top. Quantum Electron. 18(2), 996–1005 (2012).
[Crossref]

Muriel, M. A.

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]

J. Azana and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).

M. A. Muriel, J. Azaña, and A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. 24(1), 1–3 (1999).
[Crossref] [PubMed]

J. Azaña and M. A. Muriel, “Temporal Talbot effect in fiber gratings and its applications,” Appl. Opt. 38(32), 6700–6704 (1999).
[Crossref] [PubMed]

Nazarathy, M.

Nuccio, S. R.

A. E. Willner, O. F. Yilmaz, J. A. Wang, X. X. Wu, A. Bogoni, L. Zhang, and S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17(2), 320–332 (2011).
[Crossref]

Oxenlowe, L. K.

L. K. Oxenlowe, H. Ji, M. Galili, M. H. Pu, H. Hu, H. C. H. Mulvad, K. Yvind, J. M. Hvam, A. T. Clausen, and P. Jeppesen, “Silicon photonics for signal processing of Tbit/s serial data signals,” IEEE J. Sel. Top. Quantum Electron. 18(2), 996–1005 (2012).
[Crossref]

Ozdur, I.

P. J. Delfyett, I. Ozdur, N. Hoghooghi, M. Akbulut, J. Davila-Rodriguez, and S. Bhooplapur, “Advanced ultrafast technologies based on optical frequency combs,” IEEE J. Sel. Top. Quantum Electron. 18(1), 258–274 (2012).
[Crossref]

Park, Y.

Poti, L.

Pritchard, D. E.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[Crossref] [PubMed]

Pu, M. H.

L. K. Oxenlowe, H. Ji, M. Galili, M. H. Pu, H. Hu, H. C. H. Mulvad, K. Yvind, J. M. Hvam, A. T. Clausen, and P. Jeppesen, “Silicon photonics for signal processing of Tbit/s serial data signals,” IEEE J. Sel. Top. Quantum Electron. 18(2), 996–1005 (2012).
[Crossref]

Pudo, D.

Rayleigh, L.

L. Rayleigh, “On copying diffraction gratings and on some phenomenon connected therewith,” Philos. Mag. 11(67), 196–205 (1881).
[Crossref]

Refregier, P.

Saffman, M.

A. V. Mamaev and M. Saffman, “Selection of unstable patterns and control of optical turbulence by Fourier plane filtering,” Phys. Rev. Lett. 80(16), 3499–3502 (1998).
[Crossref]

Salem, R.

Schmiedmayer, J.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[Crossref] [PubMed]

Scott, R. P.

S. J. B. Yoo, R. P. Scott, D. J. Geisler, N. K. Fontaine, and F. M. Soares, “Terahertz information and signal processing by RF-photonics,” IEEE Tran. Terahertz Sci. Technol. 2(2), 167–176 (2012).
[Crossref]

Soares, F. M.

S. J. B. Yoo, R. P. Scott, D. J. Geisler, N. K. Fontaine, and F. M. Soares, “Terahertz information and signal processing by RF-photonics,” IEEE Tran. Terahertz Sci. Technol. 2(2), 167–176 (2012).
[Crossref]

Sukhoruk, A. P.

S. A. Akhmanov, A. P. Sukhoruk, and A. S. Chirkin, “Nonstationary phenomena and space-time analogy in nonlinear optics,” Sov. Phys. JETP-USSR 28, 748 (1969).

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science no. IV,” Philos. Mag. 9, 401–407 (1836).

Tannian, B. E.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[Crossref] [PubMed]

Thomas, S.

Treacy, E. B.

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5(9), 454–458 (1969).
[Crossref]

Tucker, R. S.

R. S. Tucker and K. Hinton, “Energy consumption and energy density in optical and electronic signal processing,” IEEE Photonics J. 3(5), 821–833 (2011).
[Crossref]

Turner, A. C.

Turner-Foster, A. C.

van Howe, J.

Wang, C.

Wang, J. A.

A. E. Willner, O. F. Yilmaz, J. A. Wang, X. X. Wu, A. Bogoni, L. Zhang, and S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17(2), 320–332 (2011).
[Crossref]

Wehinger, S.

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[Crossref] [PubMed]

Weiner, A. M.

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]

A. M. Weiner and A. M. Kan’an, “Femtosecond pulse shaping for synthesis, processing, and time-to-space conversion of ultrafast optical waveforms,” IEEE J. Sel. Top. Quantum Electron. 4(2), 317–331 (1998).
[Crossref]

A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19(3), 161–237 (1995).
[Crossref]

Wen, J.

Willner, A. E.

A. E. Willner, O. F. Yilmaz, J. A. Wang, X. X. Wu, A. Bogoni, L. Zhang, and S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17(2), 320–332 (2011).
[Crossref]

Wu, X. X.

A. E. Willner, O. F. Yilmaz, J. A. Wang, X. X. Wu, A. Bogoni, L. Zhang, and S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17(2), 320–332 (2011).
[Crossref]

Xiao, M.

Xu, C.

Yao, J.

H. Chi and J. Yao, “All-fiber chirped microwave pulses generation based on spectral shaping and wavelength-to-time conversion,” IEEE Trans. Microwave Theory 55(9), 1958–1963 (2007).
[Crossref]

Yao, J. P.

Yilmaz, O. F.

A. E. Willner, O. F. Yilmaz, J. A. Wang, X. X. Wu, A. Bogoni, L. Zhang, and S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17(2), 320–332 (2011).
[Crossref]

Yoo, S. J. B.

S. J. B. Yoo, R. P. Scott, D. J. Geisler, N. K. Fontaine, and F. M. Soares, “Terahertz information and signal processing by RF-photonics,” IEEE Tran. Terahertz Sci. Technol. 2(2), 167–176 (2012).
[Crossref]

Yvind, K.

L. K. Oxenlowe, H. Ji, M. Galili, M. H. Pu, H. Hu, H. C. H. Mulvad, K. Yvind, J. M. Hvam, A. T. Clausen, and P. Jeppesen, “Silicon photonics for signal processing of Tbit/s serial data signals,” IEEE J. Sel. Top. Quantum Electron. 18(2), 996–1005 (2012).
[Crossref]

Zhang, L.

A. E. Willner, O. F. Yilmaz, J. A. Wang, X. X. Wu, A. Bogoni, L. Zhang, and S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17(2), 320–332 (2011).
[Crossref]

Zhang, Y.

Adv. Opt. Photon. (2)

Appl. Opt. (1)

Electron. Lett. (1)

J. Azaña, “Design specifications of time-domain spectral shaping optical system based on dispersion and temporal modulation,” Electron. Lett. 39(21), 1530–1532 (2003).
[Crossref]

IEEE J. Quantum Electron. (5)

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. 5(9), 454–458 (1969).
[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).

C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging - Part II: System performance,” IEEE J. Quantum Electron. 36, 649–655 (2000).

B. H. Kolner, “Space-time duality and the theory of temporal imaging,” IEEE J. Quantum Electron. 30(8), 1951–1963 (1994).
[Crossref]

J. Azana and M. A. Muriel, “Real-time optical spectrum analysis based on the time-space duality in chirped fiber gratings,” IEEE J. Quantum Electron. 36, 517–526 (2000).

IEEE J. Sel. Top. Quantum Electron. (5)

P. J. Delfyett, I. Ozdur, N. Hoghooghi, M. Akbulut, J. Davila-Rodriguez, and S. Bhooplapur, “Advanced ultrafast technologies based on optical frequency combs,” IEEE J. Sel. Top. Quantum Electron. 18(1), 258–274 (2012).
[Crossref]

A. E. Willner, O. F. Yilmaz, J. A. Wang, X. X. Wu, A. Bogoni, L. Zhang, and S. R. Nuccio, “Optically efficient nonlinear signal processing,” IEEE J. Sel. Top. Quantum Electron. 17(2), 320–332 (2011).
[Crossref]

L. K. Oxenlowe, H. Ji, M. Galili, M. H. Pu, H. Hu, H. C. H. Mulvad, K. Yvind, J. M. Hvam, A. T. Clausen, and P. Jeppesen, “Silicon photonics for signal processing of Tbit/s serial data signals,” IEEE J. Sel. Top. Quantum Electron. 18(2), 996–1005 (2012).
[Crossref]

A. M. Weiner and A. M. Kan’an, “Femtosecond pulse shaping for synthesis, processing, and time-to-space conversion of ultrafast optical waveforms,” IEEE J. Sel. Top. Quantum Electron. 4(2), 317–331 (1998).
[Crossref]

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]

IEEE Photon. Technol. Lett. (1)

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(3), 882–884 (2004).
[Crossref]

IEEE Photonics J. (1)

R. S. Tucker and K. Hinton, “Energy consumption and energy density in optical and electronic signal processing,” IEEE Photonics J. 3(5), 821–833 (2011).
[Crossref]

IEEE Tran. Terahertz Sci. Technol. (1)

S. J. B. Yoo, R. P. Scott, D. J. Geisler, N. K. Fontaine, and F. M. Soares, “Terahertz information and signal processing by RF-photonics,” IEEE Tran. Terahertz Sci. Technol. 2(2), 167–176 (2012).
[Crossref]

IEEE Trans. Microwave Theory (1)

H. Chi and J. Yao, “All-fiber chirped microwave pulses generation based on spectral shaping and wavelength-to-time conversion,” IEEE Trans. Microwave Theory 55(9), 1958–1963 (2007).
[Crossref]

J. Lightwave Technol. (4)

J. Mod. Opt. (1)

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
[Crossref]

J. Opt. Soc. Am. (1)

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

Nat. Photonics (1)

K. Goda and B. Jalali, “Dispersive Fourier transformation for fast continuous single-shot measurements,” Nat. Photonics 7(2), 102–112 (2013).
[Crossref]

Opt. Commun. (1)

J. Caraquitena and J. Martí, “High-rate pulse-train generation by phase-only filtering of an electrooptic frequency comb: Analysis and optimization,” Opt. Commun. 282(18), 3686–3692 (2009).
[Crossref]

Opt. Express (2)

Opt. Lett. (5)

Opt. Photon. News (1)

M. A. Foster, R. Salem, and A. L. Gaeta, “Ultrahigh-speed optical processing using space-time duality,” Opt. Photon. News 22(5), 29–35 (2011).
[Crossref]

Philos. Mag. (2)

H. F. Talbot, “Facts relating to optical science no. IV,” Philos. Mag. 9, 401–407 (1836).

L. Rayleigh, “On copying diffraction gratings and on some phenomenon connected therewith,” Philos. Mag. 11(67), 196–205 (1881).
[Crossref]

Phys. Rev. A (1)

M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, “Near-field imaging of atom diffraction gratings: the atomic Talbot effect,” Phys. Rev. A 51(1), R14–R17 (1995).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

A. V. Mamaev and M. Saffman, “Selection of unstable patterns and control of optical turbulence by Fourier plane filtering,” Phys. Rev. Lett. 80(16), 3499–3502 (1998).
[Crossref]

Prog. Quantum Electron. (1)

A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quantum Electron. 19(3), 161–237 (1995).
[Crossref]

Sov. Phys. JETP-USSR (1)

S. A. Akhmanov, A. P. Sukhoruk, and A. S. Chirkin, “Nonstationary phenomena and space-time analogy in nonlinear optics,” Sov. Phys. JETP-USSR 28, 748 (1969).

Other (2)

J. W. Goodman, Introduction to Fourier Optics (Roberts and Co., 2005).

A. Papoulis, Systems and Transforms With Applications in Optics (McGraw-Hill, 1968).

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Figures (9)

Fig. 1
Fig. 1 (a) Train of pulses that have been intensity modulated by a set of coefficients with N = 4 and (b) grouping of the different trains depending on the modulating coefficient.
Fig. 2
Fig. 2 Output of the system, for the input train shown in Fig. 1, when (a) s is even or (b) s is odd.
Fig. 3
Fig. 3 (a) Train of pulses modulated at the input of the system with N = 3, and (b) at the output of the system.
Fig. 4
Fig. 4 Proposed setup with intensity and phase modulation in the time domain followed by a dispersive device (EO-IM: Electrooptic Intensity Modulator, EO-PM: Electrooptic Phase Modulator, DCF: Dispersion Compensating Fibre).
Fig. 5
Fig. 5 Train of pulses at the input (a) and at the output (b) of the system when no modulation is applied.
Fig. 6
Fig. 6 Train of pulses at the input (dashed) and output (solid) of the system and the expected output envelope (dotted), when intensity-only coefficients (a) 1000000000 and (b) 1000000001 are applied.
Fig. 7
Fig. 7 Train of pulses at the input (dashed) and output (solid) of the system and the expected output envelope (dotted) for a (a) ramp, (b) triangular, (c) binary data and (d) random envelope.
Fig. 8
Fig. 8 Train of pulses at the output of the system for a desired triangular envelope with N = 20; 20 periods of the output are shown in (a) and zooms of the signal in (a) for three specific periods are shown in (b) to (d).
Fig. 9
Fig. 9 Train of pulses at the output of the system for a desired triangular envelope with N = 60; 60 periods of the output are shown in (a) and zooms of the signal in (a) for three specific periods are shown in (b) to (d).

Equations (47)

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

x( t )=c( t ) k= a( tk T 0 )
c(t)=c(t+N T 0 )
c(t)c(k T 0 )= c k for k T 0 Δt 2 tk T 0 + Δt 2 where k is an integer
x( t )= l=0 N1 [ c l k= a( tl T 0 k T 1 ) ]
| ϕ |= T 1 2 s 2π = N 2 T 0 2 s 2π s=±1, ±2, ±3...
y( t )= l=0 N1 [ c l k= a( tk T 1 l T 0 ) ]
P out ( t )= l=0 N1 | c l | 2 k= | a( tk T 1 l T 0 ) | 2
y( t )= l=0 N1 [ c l k= a( tk T 1 l T 0 T 1 /2 ) ]
P out ( t )= l=0 N1 | c l | 2 k= | a( tk T 1 l T 0 T 1 /2 ) | 2
| ϕ |= T 1 2 2π s m = N 2 T 0 2 2π s m { s=±1, ±2, ±3... m= 2, 3,4...
Δt T 0 m = T 1 mN
c l k= A k a( tk T 1 m l T 0 )
A k = 1 m q=0 m1 exp( jπ{ s m q 2 + 2k m q } )
y( t )= k= l=0 N1 c l A k a( tk N m T 0 l T 0 )
c l k= B k a( tk T 1 m l T 0 T 1 2m )
B k = 1 m q=0 m1 exp( jπ{ s m q 2 + 2k+1 m q } )
y( t )= k= l=0 N1 c l B k a( tk N m T 0 l T 0 N 2m T 0 )
y( t )= k'= [ l=0 N1 A k ',l c l ]a( tk' T 0 N ) = = k'= n k a( tk' T 0 N )
n k' = 1 N 2 l=0 N1 q=0 N 2 1 c l exp( j π N 2 { q 2 +( 2k'2lN )q } ) = = 1 N 2 q=0 N 2 1 [ l=0 N1 c l exp( j 2π N lq ) ]exp( j π N 2 { q 2 +2k'q } )
n k' = 1 N 2 w=0 N1 [ z=0 N1 [ l=0 N1 c l exp( j π N 2 { ( w+zN ) 2 +( 2k'2lN )( w+zN ) } ) ] ] = 1 N 2 w=0 N1 [ exp( j π N 2 { w 2 +2k'w } ) l=0 N1 [ c l exp( j 2π N lw ) ] z=0 N1 [ ( 1 ) z exp( j 2π N z( w+k' ) ) ] ]
n k' = 1 N 2 w=0 N1 [ C w exp( j π N 2 { w 2 +2wk' } ) z=0 N1 [ ( 1 ) z exp( j 2π N z{ w+k' } ) ] ]
C w = l=0 N1 c l exp( j 2π N lw )
z=0 N1 ( 1 ) z exp( j 2π N z{ w+k' } ) ={ 0 if wrNk'+ N 2 N if w=rNk'+ N 2
k'N/2 N r k'+N/2 1 N
w 0 = r 0 N+ N 2 k'
n k' = 1 N C w 0 exp( j π N 2 { w 0 2 +2 w 0 k' } )= = 1 N C N 2 k' exp( j π 4 )exp( jπ ( k' N ) 2 )
y( t )= k'= 1 N C N 2 k' exp( j π 4 )exp( jπ ( k' N ) 2 )a( tk' T 0 N )
P out ( t )= 1 N 2 k''= | C k'' | 2 | a( t+k'' T 0 N T 0 2 ) | 2
y( t )= k'= [ l=0 N1 B k ',l c l ]a( tk' T 0 N T 0 2N ) = = k'= m k a( tk' T 0 N T 0 2N )
m k' = 1 N 2 l=0 N1 q=0 N 2 1 c l exp( j π N 2 { q 2 +( 2k'2lN+1 )q } ) = = 1 N 2 q=0 N 2 1 [ l=0 N1 c l exp( j 2π N lq ) ]exp( j π N 2 { q 2 +( 2k'+1 )q } )
m k' = 1 N 2 w=0 N1 [ C w exp( j π N 2 { w 2 +w( 2k'+1 ) } )[ z=0 N1 ( 1 ) z exp( j π N z( 2w+2k'+1 ) ) ] ]
z=0 N1 ( 1 ) z exp( j π N z( 2w+2k'+1 ) ) ={ 0 if wrNk'+ N1 2 N if w=rNk'+ N1 2
k' ( N1 ) /2 N r k'+ ( N1 ) /2 N
m k' = 1 N C w 0 exp( j π N 2 { w 0 2 + w 0 ( 2k'+1 ) } )= = 1 N C N1 2 k' exp( j π 4 )exp( j π N 2 ( k' 1 2 ) 2 )
P out ( t )= k'= | m k' | 2 | a( tk' T 0 N T 0 2N ) | 2 = = 1 N 2 k'= | C N1 2 k' | 2 | a( t k T 0 N T 0 2N ) | 2
P out ( t )= 1 N 2 k'''= | C k''' | 2 | a( t+k''' T 0 N T 0 2 ) | 2
h=0 N1 ( 1 ) h exp( j π N hs' ) ={ 0 s'(2r+1)N N s'=(2r+1)N
h=0 N1 ( 1 ) h exp( j π N hs' ) =[ z=0 N/21 exp( j 2π N zs' ) ][ 1exp( j π N s' ) ]
z=0 M1 a z = 1 a M 1a
z=0 N/21 exp( j 2π N zs' ) = 1exp( jπs' ) [ 1+exp( j π N s' ) ][ 1exp( j π N s' ) ]
h=0 N1 ( 1 ) h exp( j π N hs' ) = 1exp( jπs' ) 1+exp( j π N s' )
h=0 N1 ( 1 ) h exp( j π N hs' ) =[ z=0 N1 2 1 exp( j π N 2zs' ) ][ 1exp( j π N s' ) ]+exp( jπ ( N1 ) N s' )
z=0 N1 2 1 exp( j π N 2zs' ) = 1exp( jπ N1 N s' ) [ 1+exp( j π N s' ) ][ 1exp( j π N s' ) ]
h=0 N1 ( 1 ) h exp( j π N hs' ) = 1+exp( jπs' ) 1+exp( j π N s' )
h=0 N1 ( 1 ) h exp( j π N hs' ) = 1 ( 1 ) ( N+s' ) 1+exp( j π N s' )
lim s'(2r+1)N 1 ( 1 ) N+s' 1+exp( j π N s' ) = lim s'(2r+1)N jπ ( 1 ) N+s' j π N exp( j π N s' ) =N
h=0 N1 ( 1 ) h exp( j π N hs' ) ={ 0 s'(2r+1)N N s'=(2r+1)N

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