Abstract

We propose what we believe to be a new approach to correct the residual aberrations of time lenses implemented through electro-optic sinusoidal phase modulation. The method is based on a single modulator driven with various harmonic channels coming from the same clock signal. Proper weighting of the amplitude and phase of the channels allows us to achieve nearly perfect parabolic phase modulation over a fraction of the clock period. We show numerically that our proposal only requires three harmonics of the clock frequency to achieve truly parabolic modulation over a time window that extends across 70% of the period. We illustrate the benefits of this aberration-free scheme in a pulse compression scenario and compare its performance with previous conventional Fourier optimization techniques. Our study offers simple guidelines to achieve indeed quadratic phase modulation in situations where electro-optic phase modulation in lithium niobate is required, such as integrated frequency comb technology or high-speed optical pulse shaping.

© 2010 Optical Society of America

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References

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  1. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, 1991).
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    [CrossRef]
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    [CrossRef]
  4. B. H. Kolner, “Active pulse compression using an integrated electro-optic phase modulator,” Appl. Phys. Lett. 52, 1122–1124 (1988).
    [CrossRef]
  5. C. V. Bennett, R. P. Scott, and B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical-data with an up-conversion time microscope,” Appl. Phys. Lett. 65, 2513–2515 (1994).
    [CrossRef]
  6. B. H. Kolner and M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14, 630–632 (1989).
    [CrossRef] [PubMed]
  7. M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–273 (1994).
    [CrossRef]
  8. M. Nakazawa, T. Hirooka, F. Futami, and S. Watanabe, “Ideal distortion-free transmission using optical Fourier transformation and Fourier transform-limited optical pulses,” IEEE Photon. Technol. Lett. 16, 1059–1061 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  13. C. V. Bennett and B. H. Kolner, “Principles of parametric temporal imaging I: system configurations,” IEEE J. Quantum Electron. 36, 430–437 (2000).
    [CrossRef]
  14. C. V. Bennett, B. D. Moran, C. Langrock, M. M. Fejer, and M. Ibsen, “640 GHz real-time recording using temporal imaging,” in Conference on Lasers and Electro-Optics (CLEO) (2008), paper CTuA6.
  15. L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
    [CrossRef]
  16. T. Hirooka and M. Nakazawa, “All-optical 40 GHz time-domain Fourier transformation using XPM with a dark parabolic pulse,” IEEE Photon. Technol. Lett. 20, 1869–1871 (2008).
    [CrossRef]
  17. R. Salem, M. A. Foster, A. C. Turner, D. F. Geraghty, M. Lipson, and A. L. Gaeta, “Optical time lens based on four-wave mixing on a silicon chip,” Opt. Lett. 33, 1047–1049 (2008).
    [CrossRef] [PubMed]
  18. M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Silicon-chip-based ultrafast optical oscilloscope,” Nature 456, 81–84 (2008).
    [CrossRef] [PubMed]
  19. Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “High-resolution spectroscopy using a frequency magnifier,” Opt. Express 17, 5691–5697 (2009).
    [CrossRef] [PubMed]
  20. 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, 4324–4329 (2009).
    [CrossRef] [PubMed]
  21. M. A. Foster, R. Salem, Y. Okawachi, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, “Ultrafast waveform compression using a time-domain telescope,” Nat. Photonics 3, 581–585 (2009).
    [CrossRef]
  22. V. Torres-Company, J. Lancis, and P. Andrés, “Lossless equalization of frequency combs,” Opt. Lett. 33, 1822–1824 (2008).
    [CrossRef] [PubMed]
  23. T. Yamamoto, T. Komukai, K. Suzuki, and A. Takada, “Multicarrier light source with flattened spectrum using phase modulators and dispersion medium,” J. Lightwave Technol. 27, 4297–4305 (2009).
    [CrossRef]
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    [CrossRef]
  25. Y. Liao, H. J. Zhou, and Z. Meng, “Modulation efficiency of a LiNbO3 waveguide electro-optic intensity modulator operating at high microwave efficiency,” Opt. Lett. 34, 1822–1824 (2009).
    [CrossRef] [PubMed]
  26. M. D. Pelusi, Y. Matsui, and A. Suzuki, “Electrooptic phase modulation of stretched 250-fs pulses for suppression of third-order fiber dispersion in transmission,” IEEE Photon. Technol. Lett. 11, 1461–1463 (1999).
    [CrossRef]
  27. M. Haner and W. S. Warren, “Synthesis of crafted optical pulses by time domain modulation in a fiber-grating compressor,” Appl. Phys. Lett. 52, 1458–1461 (1988).
    [CrossRef]

2009 (6)

2008 (4)

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

V. Torres-Company, J. Lancis, and P. Andrés, “Lossless equalization of frequency combs,” Opt. Lett. 33, 1822–1824 (2008).
[CrossRef] [PubMed]

T. Hirooka and M. Nakazawa, “All-optical 40 GHz time-domain Fourier transformation using XPM with a dark parabolic pulse,” IEEE Photon. Technol. Lett. 20, 1869–1871 (2008).
[CrossRef]

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

2006 (1)

2004 (2)

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

M. Nakazawa, T. Hirooka, F. Futami, and S. Watanabe, “Ideal distortion-free transmission using optical Fourier transformation and Fourier transform-limited optical pulses,” IEEE Photon. Technol. Lett. 16, 1059–1061 (2004).
[CrossRef]

2003 (1)

2001 (1)

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

2000 (2)

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

L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
[CrossRef]

1999 (1)

M. D. Pelusi, Y. Matsui, and A. Suzuki, “Electrooptic phase modulation of stretched 250-fs pulses for suppression of third-order fiber dispersion in transmission,” IEEE Photon. Technol. Lett. 11, 1461–1463 (1999).
[CrossRef]

1994 (3)

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

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

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–273 (1994).
[CrossRef]

1989 (1)

1988 (2)

M. Haner and W. S. Warren, “Synthesis of crafted optical pulses by time domain modulation in a fiber-grating compressor,” Appl. Phys. Lett. 52, 1458–1461 (1988).
[CrossRef]

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

1983 (1)

Akhmanov, S. A.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, 1991).

Andrés, P.

Banyai, W. C.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–273 (1994).
[CrossRef]

Barthélémy, A.

L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
[CrossRef]

Bennett, C. V.

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

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

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

C. V. Bennett, B. D. Moran, C. Langrock, M. M. Fejer, and M. Ibsen, “640 GHz real-time recording using temporal imaging,” in Conference on Lasers and Electro-Optics (CLEO) (2008), paper CTuA6.

Bloom, D. M.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–273 (1994).
[CrossRef]

Chen, L. R.

Chirkin, A. S.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, 1991).

Fejer, M. M.

C. V. Bennett, B. D. Moran, C. Langrock, M. M. Fejer, and M. Ibsen, “640 GHz real-time recording using temporal imaging,” in Conference on Lasers and Electro-Optics (CLEO) (2008), paper CTuA6.

Foster, M. A.

Froehly, C.

L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
[CrossRef]

Futami, F.

M. Nakazawa, T. Hirooka, F. Futami, and S. Watanabe, “Ideal distortion-free transmission using optical Fourier transformation and Fourier transform-limited optical pulses,” IEEE Photon. Technol. Lett. 16, 1059–1061 (2004).
[CrossRef]

Gaeta, A. L.

Geraghty, D. F.

Godil, A. A.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–273 (1994).
[CrossRef]

Haner, M.

M. Haner and W. S. Warren, “Synthesis of crafted optical pulses by time domain modulation in a fiber-grating compressor,” Appl. Phys. Lett. 52, 1458–1461 (1988).
[CrossRef]

Hansryd, J.

Hirooka, T.

T. Hirooka and M. Nakazawa, “All-optical 40 GHz time-domain Fourier transformation using XPM with a dark parabolic pulse,” IEEE Photon. Technol. Lett. 20, 1869–1871 (2008).
[CrossRef]

M. Nakazawa, T. Hirooka, F. Futami, and S. Watanabe, “Ideal distortion-free transmission using optical Fourier transformation and Fourier transform-limited optical pulses,” IEEE Photon. Technol. Lett. 16, 1059–1061 (2004).
[CrossRef]

Ibsen, M.

C. V. Bennett, B. D. Moran, C. Langrock, M. M. Fejer, and M. Ibsen, “640 GHz real-time recording using temporal imaging,” in Conference on Lasers and Electro-Optics (CLEO) (2008), paper CTuA6.

Jannson, T.

Kauffman, M. T.

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–273 (1994).
[CrossRef]

Kolner, B. H.

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

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

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

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

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

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

Komukai, T.

Lancis, J.

Langrock, C.

C. V. Bennett, B. D. Moran, C. Langrock, M. M. Fejer, and M. Ibsen, “640 GHz real-time recording using temporal imaging,” in Conference on Lasers and Electro-Optics (CLEO) (2008), paper CTuA6.

Liao, Y.

Lipson, M.

Louradour, F.

L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
[CrossRef]

Matsui, Y.

M. D. Pelusi, Y. Matsui, and A. Suzuki, “Electrooptic phase modulation of stretched 250-fs pulses for suppression of third-order fiber dispersion in transmission,” IEEE Photon. Technol. Lett. 11, 1461–1463 (1999).
[CrossRef]

Meng, Z.

Messager, V.

L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
[CrossRef]

Moran, B. D.

C. V. Bennett, B. D. Moran, C. Langrock, M. M. Fejer, and M. Ibsen, “640 GHz real-time recording using temporal imaging,” in Conference on Lasers and Electro-Optics (CLEO) (2008), paper CTuA6.

Mouradian, L. K.

L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
[CrossRef]

Nakazawa, M.

T. Hirooka and M. Nakazawa, “All-optical 40 GHz time-domain Fourier transformation using XPM with a dark parabolic pulse,” IEEE Photon. Technol. Lett. 20, 1869–1871 (2008).
[CrossRef]

M. Nakazawa, T. Hirooka, F. Futami, and S. Watanabe, “Ideal distortion-free transmission using optical Fourier transformation and Fourier transform-limited optical pulses,” IEEE Photon. Technol. Lett. 16, 1059–1061 (2004).
[CrossRef]

Nazarathy, M.

Okawachi, Y.

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

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

Pelusi, M. D.

M. D. Pelusi, Y. Matsui, and A. Suzuki, “Electrooptic phase modulation of stretched 250-fs pulses for suppression of third-order fiber dispersion in transmission,” IEEE Photon. Technol. Lett. 11, 1461–1463 (1999).
[CrossRef]

Salem, R.

Scott, R. P.

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

Shi, Y. Q.

Suzuki, A.

M. D. Pelusi, Y. Matsui, and A. Suzuki, “Electrooptic phase modulation of stretched 250-fs pulses for suppression of third-order fiber dispersion in transmission,” IEEE Photon. Technol. Lett. 11, 1461–1463 (1999).
[CrossRef]

Suzuki, K.

Takada, A.

Torres-Company, V.

Turner, A. C.

Turner-Foster, A. C.

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

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

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, 4324–4329 (2009).
[CrossRef] [PubMed]

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

van Howe, J.

Vysloukh, V. A.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, 1991).

Warren, W. S.

M. Haner and W. S. Warren, “Synthesis of crafted optical pulses by time domain modulation in a fiber-grating compressor,” Appl. Phys. Lett. 52, 1458–1461 (1988).
[CrossRef]

Watanabe, S.

M. Nakazawa, T. Hirooka, F. Futami, and S. Watanabe, “Ideal distortion-free transmission using optical Fourier transformation and Fourier transform-limited optical pulses,” IEEE Photon. Technol. Lett. 16, 1059–1061 (2004).
[CrossRef]

Willner, A. E.

Xu, C.

Yamamoto, T.

Yan, L. S.

Zhou, H. J.

Appl. Phys. Lett. (4)

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

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

M. T. Kauffman, W. C. Banyai, A. A. Godil, and D. M. Bloom, “Time-to-frequency converter for measuring picosecond optical pulses,” Appl. Phys. Lett. 64, 270–273 (1994).
[CrossRef]

M. Haner and W. S. Warren, “Synthesis of crafted optical pulses by time domain modulation in a fiber-grating compressor,” Appl. Phys. Lett. 52, 1458–1461 (1988).
[CrossRef]

IEEE J. Quantum Electron. (4)

L. K. Mouradian, F. Louradour, V. Messager, A. Barthélémy, and C. Froehly, “Spectro-temporal imaging of femtosecond events,” IEEE J. Quantum Electron. 36, 795–801 (2000).
[CrossRef]

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

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

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

IEEE Photon. Technol. Lett. (3)

M. Nakazawa, T. Hirooka, F. Futami, and S. Watanabe, “Ideal distortion-free transmission using optical Fourier transformation and Fourier transform-limited optical pulses,” IEEE Photon. Technol. Lett. 16, 1059–1061 (2004).
[CrossRef]

T. Hirooka and M. Nakazawa, “All-optical 40 GHz time-domain Fourier transformation using XPM with a dark parabolic pulse,” IEEE Photon. Technol. Lett. 20, 1869–1871 (2008).
[CrossRef]

M. D. Pelusi, Y. Matsui, and A. Suzuki, “Electrooptic phase modulation of stretched 250-fs pulses for suppression of third-order fiber dispersion in transmission,” IEEE Photon. Technol. Lett. 11, 1461–1463 (1999).
[CrossRef]

J. Lightwave Technol. (3)

Nat. Photonics (1)

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

Nature (1)

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

Opt. Express (3)

Opt. Lett. (6)

Other (2)

C. V. Bennett, B. D. Moran, C. Langrock, M. M. Fejer, and M. Ibsen, “640 GHz real-time recording using temporal imaging,” in Conference on Lasers and Electro-Optics (CLEO) (2008), paper CTuA6.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, Optics of Femtosecond Laser Pulses (American Institute of Physics, 1991).

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Figures (5)

Fig. 1
Fig. 1

Schematic setup for the implementation of the nearly aberration-free time lens. The input periodic optical signal acquires a fully undistorted quadratic phase modulation (dashed line) while preserving the intensity profile (solid line). Since the optical phase acquired by propagating through the EOPM is proportional to the electrical signal driving it, to achieve this goal we have to synthesize properly the electrical signal. This is done by delaying, weighting (with variable electrical amplifiers), and summing a common clock signal.

Fig. 2
Fig. 2

Plot of the values of the parameter A q in terms of the fraction coefficient α to reproduce the parabolic profile 2 Δ θ π 2 t 2 / T 2 with only the first three terms of the harmonic series, whose fundamental period is T.

Fig. 3
Fig. 3

(a) Comparison among the lens profiles corresponding to single-cosine approximation (long-dashed curve), Fourier expansion (dashed-dotted curve), NLSS fit with coefficients provided by Eq. (4) (solid curve), and ideal parabolic profile (dotted curve); (b) relative deviation between the parabolic profile and the above three curves.

Fig. 4
Fig. 4

Left and right columns show results for pulse compression results throughout EOPM lensing and fiber propagation for Δ θ = π   rad and Δ θ = 2.5 π   rad , respectively. The curve codes are the same as those in Fig. 3. The initial pulse durations are (a) 38, (b) 67, and (c) 86 ps.

Fig. 5
Fig. 5

Same as in Fig. 4 but for V 0 = 2 V π (left column) and V 0 = 8 V π (right column).

Equations (8)

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

ϕ ( t ) = π V 0 2 V π cos ( 2 π t T ) = Δ θ   cos ( 2 π t T ) 2 Δ θ π 2 t 2 T 2 ,
φ ( t ) = Δ θ q = 0 n A q   cos ( q 2 π t T ) ,
S = i = 1 N ( 2 π 2 t i 2 T 2 q = 0 n A q   cos ( q 2 π t i T ) ) 2
A 1 = 1.5 + 0.54 α 2 ,
A 2 = 0.125 0.0176 e 3.19 α ,
A 3 = 0.011 + 0.00062 e 6.07 α .
D ( t ) = f ( t ) + 2 Δ θ π 2 t 2 T 2 ,
Ω ( α ) = 1 T T / 2 + T / 2 | Ψ ( t ) ( q = 0 n A q ( α ) cos ( q 2 π t T ) + 2 π 2 t 2 T ) | d t .

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