Abstract

The frequency-to-time mapping technique (also known as the temporal far-field phenomenon) usually requires a significant amount of dispersion to stretch an ultrashort optical pulse so that the intensity profile becomes a scaled replica of its optical spectrum. In this work, we study the near-to-far-field transition and find that the far-field condition can be relaxed in some cases relevant for radio-frequency (RF) waveform generation. This observation has allowed us to achieve intensity signals with an ultrabroad RF bandwidth content.

© 2011 OSA

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References

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  1. A. M. Weiner, Ultrafast Optics, (Wiley Interscience, 2009).
  2. V. Torres-Company, J. Lancis, and P. Andrés, “Space-Time analogies in Optics,” Prog. Opt.in press).
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    [CrossRef]
  4. 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]
  5. D. R. Solli, J. Chou, and B. Jalali, “Amplified wavelength-time transformation for real-time spectroscopy,” Nat. Photonics 2(1), 48–51 (2008).
    [CrossRef]
  6. M. H. Khan, H. Shen, Y. Xuan, L. Zhao, S. J. Xiao, D. E. Leaird, A. M. Weiner, and M. Qi, “Ultrabroad-bandwidth arbitrary radiofrequency waveform generation with a silicon photonic chip-based spectral shaper,” Nat. Photonics 4(2), 117–122 (2010).
    [CrossRef]
  7. S. Moon and D. Y. Kim, “Ultra-high-speed optical coherence tomography with a stretched pulse supercontinuum source,” Opt. Express 14(24), 11575–11584 (2006).
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    [CrossRef] [PubMed]
  9. S. Thomas, A. Malacarne, F. Fresi, L. Poti, and J. Azaña, “Fiber-based programmable picosecond optical pulse shaper,” J. Lightwave Technol. 28(12), 1832–1843 (2010).
    [CrossRef]
  10. K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458(7242), 1145–1149 (2009).
    [CrossRef] [PubMed]
  11. D. R. Solli, S. Gupta, and B. Jalali, “Optical phase recovery in the dispersive Fourier transformation,” Appl. Phys. Lett. 95(23), 231108 (2009).
    [CrossRef]
  12. J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. 15(4), 581–583 (2003).
    [CrossRef]
  13. I. S. Lin, J. D. McKinney, and A. M. Weiner, “Photonic synthesis of broadband microwave arbitrary waveforms applicable to ultrawideband communication,” IEEE Microw. Wirel. Compon. Lett. 15(4), 226–228 (2005).
    [CrossRef]
  14. J. W. Goodman, Introduction to Fourier Optics, 3rd ed., (Roberts and Co. Publishers, 2004).
  15. W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 2nd ed., (John Wiley and Sons, 1998).
  16. C. Wang, F. Zeng, and J. P. Yao, “All-fiber ultrawideband pulse generation based on spectral-shaping and dispersion-induced frequency-to-time conversion,” IEEE Photon. Technol. Lett. 19(3), 137–139 (2007).
    [CrossRef]
  17. M. Abtahi, M. Dastmalchi, S. LaRochelle, and L. A. Rusch, “Generation of arbitrary UWB waveforms by spectral shaping and thermally controlled apodized FBGs,” J. Lightwave Technol. 27(23), 5276–5283 (2009).
    [CrossRef]
  18. J. D. McKinney, “Background-free arbitrary waveform generation via polarization pulse shaping,” IEEE Photon. Technol. Lett. 22(16), 1193–1195 (2010).
    [CrossRef]
  19. Y. Liu, S. G. Park, and A. M. Weiner, “Enhancement of narrow-band terahertz radiation from photoconducting antennas by optical pulse shaping,” Opt. Lett. 21(21), 1762–1764 (1996).
    [CrossRef] [PubMed]
  20. Y. Liu, S. G. Park, and A. M. Weiner, “Terahertz waveform synthesis via optical pulse shaping,” IEEE J. Sel. Top. Quantum Electron. 2, 709–719 (1997).
  21. H. N. Chapman and K. A. Nugent, “Coherent lensless X-ray imaging,” Nat. Photonics 4(12), 833–839 (2010).
    [CrossRef]
  22. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
    [CrossRef]
  23. J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett. 35(25), 2223–2224 (1999).
    [CrossRef]
  24. H. Chi, F. Zeng, and J. P. Yao, “Photonic generation of microwave signals based on pulse shaping,” IEEE Photon. Technol. Lett. 19(9), 668–670 (2007).
    [CrossRef]
  25. R. E. Saperstein and Y. Fainman, “Information processing with longitudinal spectral decomposition of ultrafast pulses,” Appl. Opt. 47(4), A21–A31 (2008).
    [CrossRef] [PubMed]

2010 (5)

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

M. H. Asghari, Y. Park, and J. Azaña, “Complex-field measurement of ultrafast dynamic optical waveforms based on real-time spectral interferometry,” Opt. Express 18(16), 16526–16538 (2010).
[CrossRef] [PubMed]

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

J. D. McKinney, “Background-free arbitrary waveform generation via polarization pulse shaping,” IEEE Photon. Technol. Lett. 22(16), 1193–1195 (2010).
[CrossRef]

H. N. Chapman and K. A. Nugent, “Coherent lensless X-ray imaging,” Nat. Photonics 4(12), 833–839 (2010).
[CrossRef]

2009 (3)

M. Abtahi, M. Dastmalchi, S. LaRochelle, and L. A. Rusch, “Generation of arbitrary UWB waveforms by spectral shaping and thermally controlled apodized FBGs,” J. Lightwave Technol. 27(23), 5276–5283 (2009).
[CrossRef]

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458(7242), 1145–1149 (2009).
[CrossRef] [PubMed]

D. R. Solli, S. Gupta, and B. Jalali, “Optical phase recovery in the dispersive Fourier transformation,” Appl. Phys. Lett. 95(23), 231108 (2009).
[CrossRef]

2008 (2)

D. R. Solli, J. Chou, and B. Jalali, “Amplified wavelength-time transformation for real-time spectroscopy,” Nat. Photonics 2(1), 48–51 (2008).
[CrossRef]

R. E. Saperstein and Y. Fainman, “Information processing with longitudinal spectral decomposition of ultrafast pulses,” Appl. Opt. 47(4), A21–A31 (2008).
[CrossRef] [PubMed]

2007 (2)

H. Chi, F. Zeng, and J. P. Yao, “Photonic generation of microwave signals based on pulse shaping,” IEEE Photon. Technol. Lett. 19(9), 668–670 (2007).
[CrossRef]

C. Wang, F. Zeng, and J. P. Yao, “All-fiber ultrawideband pulse generation based on spectral-shaping and dispersion-induced frequency-to-time conversion,” IEEE Photon. Technol. Lett. 19(3), 137–139 (2007).
[CrossRef]

2006 (1)

2005 (1)

I. S. Lin, J. D. McKinney, and A. M. Weiner, “Photonic synthesis of broadband microwave arbitrary waveforms applicable to ultrawideband communication,” IEEE Microw. Wirel. Compon. Lett. 15(4), 226–228 (2005).
[CrossRef]

2003 (1)

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. 15(4), 581–583 (2003).
[CrossRef]

2000 (1)

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
[CrossRef]

1999 (2)

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett. 35(25), 2223–2224 (1999).
[CrossRef]

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]

1997 (2)

Y. C. Tong, L. Y. Chan, and H. K. Tsang, “Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33(11), 983–985 (1997).
[CrossRef]

Y. Liu, S. G. Park, and A. M. Weiner, “Terahertz waveform synthesis via optical pulse shaping,” IEEE J. Sel. Top. Quantum Electron. 2, 709–719 (1997).

1996 (1)

Abtahi, M.

Andrés, P.

V. Torres-Company, J. Lancis, and P. Andrés, “Space-Time analogies in Optics,” Prog. Opt.in press).

Asghari, M. H.

Azaña, J.

Carballar, A.

Chan, L. Y.

Y. C. Tong, L. Y. Chan, and H. K. Tsang, “Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33(11), 983–985 (1997).
[CrossRef]

Chapman, H. N.

H. N. Chapman and K. A. Nugent, “Coherent lensless X-ray imaging,” Nat. Photonics 4(12), 833–839 (2010).
[CrossRef]

Chen, L. R.

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett. 35(25), 2223–2224 (1999).
[CrossRef]

Chi, H.

H. Chi, F. Zeng, and J. P. Yao, “Photonic generation of microwave signals based on pulse shaping,” IEEE Photon. Technol. Lett. 19(9), 668–670 (2007).
[CrossRef]

Chou, J.

D. R. Solli, J. Chou, and B. Jalali, “Amplified wavelength-time transformation for real-time spectroscopy,” Nat. Photonics 2(1), 48–51 (2008).
[CrossRef]

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. 15(4), 581–583 (2003).
[CrossRef]

Dastmalchi, M.

Fainman, Y.

Fresi, F.

Goda, K.

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458(7242), 1145–1149 (2009).
[CrossRef] [PubMed]

Gupta, S.

D. R. Solli, S. Gupta, and B. Jalali, “Optical phase recovery in the dispersive Fourier transformation,” Appl. Phys. Lett. 95(23), 231108 (2009).
[CrossRef]

Han, Y.

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. 15(4), 581–583 (2003).
[CrossRef]

Jalali, B.

D. R. Solli, S. Gupta, and B. Jalali, “Optical phase recovery in the dispersive Fourier transformation,” Appl. Phys. Lett. 95(23), 231108 (2009).
[CrossRef]

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458(7242), 1145–1149 (2009).
[CrossRef] [PubMed]

D. R. Solli, J. Chou, and B. Jalali, “Amplified wavelength-time transformation for real-time spectroscopy,” Nat. Photonics 2(1), 48–51 (2008).
[CrossRef]

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. 15(4), 581–583 (2003).
[CrossRef]

Khan, M. H.

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

Kim, D. Y.

Lancis, J.

V. Torres-Company, J. Lancis, and P. Andrés, “Space-Time analogies in Optics,” Prog. Opt.in press).

LaRochelle, S.

Leaird, D. E.

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

Lin, I. S.

I. S. Lin, J. D. McKinney, and A. M. Weiner, “Photonic synthesis of broadband microwave arbitrary waveforms applicable to ultrawideband communication,” IEEE Microw. Wirel. Compon. Lett. 15(4), 226–228 (2005).
[CrossRef]

Liu, Y.

Y. Liu, S. G. Park, and A. M. Weiner, “Terahertz waveform synthesis via optical pulse shaping,” IEEE J. Sel. Top. Quantum Electron. 2, 709–719 (1997).

Y. Liu, S. G. Park, and A. M. Weiner, “Enhancement of narrow-band terahertz radiation from photoconducting antennas by optical pulse shaping,” Opt. Lett. 21(21), 1762–1764 (1996).
[CrossRef] [PubMed]

Malacarne, A.

McKinney, J. D.

J. D. McKinney, “Background-free arbitrary waveform generation via polarization pulse shaping,” IEEE Photon. Technol. Lett. 22(16), 1193–1195 (2010).
[CrossRef]

I. S. Lin, J. D. McKinney, and A. M. Weiner, “Photonic synthesis of broadband microwave arbitrary waveforms applicable to ultrawideband communication,” IEEE Microw. Wirel. Compon. Lett. 15(4), 226–228 (2005).
[CrossRef]

Moon, S.

Muriel, M. A.

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, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett. 35(25), 2223–2224 (1999).
[CrossRef]

Nugent, K. A.

H. N. Chapman and K. A. Nugent, “Coherent lensless X-ray imaging,” Nat. Photonics 4(12), 833–839 (2010).
[CrossRef]

Park, S. G.

Y. Liu, S. G. Park, and A. M. Weiner, “Terahertz waveform synthesis via optical pulse shaping,” IEEE J. Sel. Top. Quantum Electron. 2, 709–719 (1997).

Y. Liu, S. G. Park, and A. M. Weiner, “Enhancement of narrow-band terahertz radiation from photoconducting antennas by optical pulse shaping,” Opt. Lett. 21(21), 1762–1764 (1996).
[CrossRef] [PubMed]

Park, Y.

Poti, L.

Qi, M.

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

Rusch, L. A.

Saperstein, R. E.

Shen, H.

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

Smith, P. W. E.

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett. 35(25), 2223–2224 (1999).
[CrossRef]

Solli, D. R.

D. R. Solli, S. Gupta, and B. Jalali, “Optical phase recovery in the dispersive Fourier transformation,” Appl. Phys. Lett. 95(23), 231108 (2009).
[CrossRef]

D. R. Solli, J. Chou, and B. Jalali, “Amplified wavelength-time transformation for real-time spectroscopy,” Nat. Photonics 2(1), 48–51 (2008).
[CrossRef]

Thomas, S.

Tong, Y. C.

Y. C. Tong, L. Y. Chan, and H. K. Tsang, “Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33(11), 983–985 (1997).
[CrossRef]

Torres-Company, V.

V. Torres-Company, J. Lancis, and P. Andrés, “Space-Time analogies in Optics,” Prog. Opt.in press).

Tsang, H. K.

Y. C. Tong, L. Y. Chan, and H. K. Tsang, “Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33(11), 983–985 (1997).
[CrossRef]

Tsia, K. K.

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458(7242), 1145–1149 (2009).
[CrossRef] [PubMed]

Wang, C.

C. Wang, F. Zeng, and J. P. Yao, “All-fiber ultrawideband pulse generation based on spectral-shaping and dispersion-induced frequency-to-time conversion,” IEEE Photon. Technol. Lett. 19(3), 137–139 (2007).
[CrossRef]

Weiner, A. M.

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

I. S. Lin, J. D. McKinney, and A. M. Weiner, “Photonic synthesis of broadband microwave arbitrary waveforms applicable to ultrawideband communication,” IEEE Microw. Wirel. Compon. Lett. 15(4), 226–228 (2005).
[CrossRef]

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
[CrossRef]

Y. Liu, S. G. Park, and A. M. Weiner, “Terahertz waveform synthesis via optical pulse shaping,” IEEE J. Sel. Top. Quantum Electron. 2, 709–719 (1997).

Y. Liu, S. G. Park, and A. M. Weiner, “Enhancement of narrow-band terahertz radiation from photoconducting antennas by optical pulse shaping,” Opt. Lett. 21(21), 1762–1764 (1996).
[CrossRef] [PubMed]

Xiao, S. J.

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

Xuan, Y.

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

Yao, J. P.

C. Wang, F. Zeng, and J. P. Yao, “All-fiber ultrawideband pulse generation based on spectral-shaping and dispersion-induced frequency-to-time conversion,” IEEE Photon. Technol. Lett. 19(3), 137–139 (2007).
[CrossRef]

H. Chi, F. Zeng, and J. P. Yao, “Photonic generation of microwave signals based on pulse shaping,” IEEE Photon. Technol. Lett. 19(9), 668–670 (2007).
[CrossRef]

Zeng, F.

H. Chi, F. Zeng, and J. P. Yao, “Photonic generation of microwave signals based on pulse shaping,” IEEE Photon. Technol. Lett. 19(9), 668–670 (2007).
[CrossRef]

C. Wang, F. Zeng, and J. P. Yao, “All-fiber ultrawideband pulse generation based on spectral-shaping and dispersion-induced frequency-to-time conversion,” IEEE Photon. Technol. Lett. 19(3), 137–139 (2007).
[CrossRef]

Zhao, L.

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

Appl. Opt. (1)

Appl. Phys. Lett. (1)

D. R. Solli, S. Gupta, and B. Jalali, “Optical phase recovery in the dispersive Fourier transformation,” Appl. Phys. Lett. 95(23), 231108 (2009).
[CrossRef]

Electron. Lett. (2)

Y. C. Tong, L. Y. Chan, and H. K. Tsang, “Fibre dispersion or pulse spectrum measurement using a sampling oscilloscope,” Electron. Lett. 33(11), 983–985 (1997).
[CrossRef]

J. Azaña, L. R. Chen, M. A. Muriel, and P. W. E. Smith, “Experimental demonstration of real-time Fourier transformation using linearly chirped fibre Bragg gratings,” Electron. Lett. 35(25), 2223–2224 (1999).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

Y. Liu, S. G. Park, and A. M. Weiner, “Terahertz waveform synthesis via optical pulse shaping,” IEEE J. Sel. Top. Quantum Electron. 2, 709–719 (1997).

IEEE Microw. Wirel. Compon. Lett. (1)

I. S. Lin, J. D. McKinney, and A. M. Weiner, “Photonic synthesis of broadband microwave arbitrary waveforms applicable to ultrawideband communication,” IEEE Microw. Wirel. Compon. Lett. 15(4), 226–228 (2005).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

C. Wang, F. Zeng, and J. P. Yao, “All-fiber ultrawideband pulse generation based on spectral-shaping and dispersion-induced frequency-to-time conversion,” IEEE Photon. Technol. Lett. 19(3), 137–139 (2007).
[CrossRef]

J. Chou, Y. Han, and B. Jalali, “Adaptive RF-photonic arbitrary waveform generator,” IEEE Photon. Technol. Lett. 15(4), 581–583 (2003).
[CrossRef]

J. D. McKinney, “Background-free arbitrary waveform generation via polarization pulse shaping,” IEEE Photon. Technol. Lett. 22(16), 1193–1195 (2010).
[CrossRef]

H. Chi, F. Zeng, and J. P. Yao, “Photonic generation of microwave signals based on pulse shaping,” IEEE Photon. Technol. Lett. 19(9), 668–670 (2007).
[CrossRef]

J. Lightwave Technol. (2)

Nat. Photonics (3)

D. R. Solli, J. Chou, and B. Jalali, “Amplified wavelength-time transformation for real-time spectroscopy,” Nat. Photonics 2(1), 48–51 (2008).
[CrossRef]

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

H. N. Chapman and K. A. Nugent, “Coherent lensless X-ray imaging,” Nat. Photonics 4(12), 833–839 (2010).
[CrossRef]

Nature (1)

K. Goda, K. K. Tsia, and B. Jalali, “Serial time-encoded amplified imaging for real-time observation of fast dynamic phenomena,” Nature 458(7242), 1145–1149 (2009).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (2)

Prog. Opt. (1)

V. Torres-Company, J. Lancis, and P. Andrés, “Space-Time analogies in Optics,” Prog. Opt.in press).

Rev. Sci. Instrum. (1)

A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000).
[CrossRef]

Other (3)

A. M. Weiner, Ultrafast Optics, (Wiley Interscience, 2009).

J. W. Goodman, Introduction to Fourier Optics, 3rd ed., (Roberts and Co. Publishers, 2004).

W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 2nd ed., (John Wiley and Sons, 1998).

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Figures (6)

Fig. 1
Fig. 1

(a) Intensity profile achieved when Φ 2 =1/ ( π υ 2 ) with υ=100 GHz and Δω=100δω (blue continuous curve) compared with the version of the scaled optical spectrum (red dashed line). (b) Zoomed version of the area highlighted by the dash circle in (a). (c) Dispersion required to achieve a 95% visibility of the dip in the temporal domain for the waveform of Eq. (4) for different spectral feature widths υ . Three different relative optical bandwidths, N=Δω/δω , are considered. The trend is compared with the minimum amount predicted form Eq. (3) (green continuous line).

Fig. 2
Fig. 2

Schematic representation of the experimental setup. A home-built erbium fiber laser with ~45 nm at full-width at half maximum (FWHM) operating at 53 MHz repetition rate is used as the laser source. The light from the upper arm is shaped with a commercial pulse shaper (Finisar WaveShaper 1000S), that has the capability to manipulate the complex spectrum with 10 GHz resolution in the whole C-band (5 THz bandwidth). It consists of a programmable liquid crystal on silicon display (LCOS) placed at the focal plane of a 4f Fourier processor working in a reflective geometry [22]. Finally, the intensity profiles of the shaped pulses are measured by a home-built intensity cross-correlation apparatus working in a non-collinear background-free geometry. In the setup, an achromatic lens focuses the signal and reference beams on a 0.5 mm thick BBO nonlinear crystal to produce second harmonic radiation. The cross-correlator has a motorized mechanical delay stage and it performs the sweep in a few seconds, depending on the temporal duration of the pulse to be measured.

Fig. 3
Fig. 3

Chirped waveform: (a) Measured (red) and target (green) optical spectrum. (b)-(e) blue solid line are the numerical intensity profiles at the corresponding GDD amounts of Φ 2 =0;0.43;0.86;1.51p s 2 , and red dashed line is the spectrum scaled. (f)-(i) measured intensity profiles at the GDD amount corresponding to the simulated profile on the left column and red-dashed line the scaled measured spectrum.

Fig. 4
Fig. 4

Cosine profile: (a) Measured (red) and target (green) optical spectrum. (b) Numerically calculated evolution of the intensity profile for different GDD amounts. Simulated intensity profiles (blue line) marked in white dashed line in the 2D plot for (c) Φ 2 =0.65p s 2 and (d) Φ 2 =3.24p s 2 . Red dashed line is the spectrum scaled. (e) and (f) measured intensity profiles at the GDD amount corresponding to the same row, and red-dashed line the scaled measured spectrum.

Fig. 5
Fig. 5

Two-pulse cosine spectrum: (a) Simulation of intensity profiles for different dispersion amounts. First column shows the simulated intensity profiles (blue line) for the dispersion amounts marked in white dashed line in (a) and compared with the spectrum scaled (red dashed line) for the dispersion amounts (b) Φ 2 =0.43p s 2 (c) Φ 2 =0.86p s 2 (d) Φ 2 =1.51p s 2 . (e)-(g) measured intensity profile at the GDD amount corresponding to the same row and red-dashed line the scaled measured spectrum. (h)-(j) calculated RF spectra from the measured intensity profiles at (e)-(g).

Fig. 6
Fig. 6

Two-pulse cosine spectrum: Evolution of the intensity profile at a fixed dispersion amount ( Φ 2 =0.43p s 2 ) and input waveform duration with different bandwidth content. (a) 0.8 THz FWHM bandwidth ; (b) 1.4 THz; and (c) 2.5 THz, leading to visibilities of 69.8%, 90.7% and 94.3%, respectively. Dashed red lines are the corresponding optical spectra scaled.

Tables (1)

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Table 1 Comparison of the Accuracy for the Different Far-field Criteria Studied

Equations (12)

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E out (t)exp( i t 2 2 Φ 2 ) E in (t')exp( i t ' 2 2 Φ 2 )exp( i tt' Φ 2 )dt' .
| Φ 2 |>> σ 0 2 2π ,
| Φ 2 |> σ 0 2 π .
E ˜ in (ω)=exp[ ω 2 / 2Δ ω 2 ]exp[ ω 2 / 2δ ω 2 ].
I out (t)= | E out (t) | 2 | g 1 (t) | 2 + | γ | 2 | g 2 (t) | 2 2| γ || g 1 (t) g 2 (t) |cos[ ( C 1 C 2 ) t 2 +ϕ ],
B max = 1 δ t min = 1 Φ 2,min 2πυ = υ 2 .
E ˜ in (ω)= 1 2 exp( ω 2 2Δ ω 2 ){ exp( i ψ 2 ω 2 )exp( iωτ /2 )+exp( iωτ /2 ) },
E ˜ in (ω)= 1 2 exp( ω 2 2Δ ω 2 ){ 1+cos( ωτ ) }.
E ˜ in (ω)= 1 2 exp( ω 2 2Δ ω 2 ){ exp( iωτ /2 )+exp( iωτ /2 ) },
| Φ 2 |> 2τ / Δω ,
| Φ 2 |> 2 σ 0 / Δω ,
r x =1 [ | E in (ω=t/ Φ 2 ) | 2 I out (t) ] 2 dt [ I out (t)] 2 dt | E in (ω=t/ Φ 2 ) | 4 dt .

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