Abstract

Pulse shaping theory is extended to include rapid waveform update for line-by-line pulse shaping. The fundamental tradeoff between response speed and waveform fidelity is illustrated by several examples.

© 2008 Optical Society of America

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References

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  1. A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Instrum. 71, 1929-1960 (2000).
    [CrossRef]
  2. S. Xiao, A. M. Weiner, and C. Lin, "Experimental and theoretical study of hyperfine WDM demulitplexer performance using the Virtually Imaged Phased-Array (VIPA)," J. Lightwave Technol. 23, 1456-1467 (2005).
    [CrossRef]
  3. S. A. Diddams, L. Hollberg, and V. Mbele, "Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb," Nature 445, 627-630 (2007).
    [CrossRef] [PubMed]
  4. Z. Jiang, D.-S. Seo, D. E. Leaird, and A. M. Weiner, "Spectral line-by-line pulse shaping," Opt. Lett. 30, 1557-1559 (2005).
    [CrossRef] [PubMed]
  5. Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, "Optical arbitrary waveform processing of more than 100 spectral comb lines," Nat. Photonics 1, 463-467 (2007).
    [CrossRef]
  6. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis," Science 288,635-639 (2000).
    [CrossRef] [PubMed]
  7. J. Ye and S. T. Cundiff, eds., Femtosecond Optical Frequency Combs Technology: Principle, Operation, and Application (Springer, New York, 2004).
  8. O. E. Martinez, "Grating and prism compressors in the case of finite beam size," J. Opt. Soc. Am. B 3, 929-934 (1986).
    [CrossRef]
  9. R. N. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, "Analysis of picoseconds pulse shape synthesis by spectral masking in a grating pulse compressor," IEEE J. Quantum Electron. 22, 682-696 (1986).
    [CrossRef]
  10. M. M. Wefers and K. A. Nelson, "Space-time profiles of shaped ultrafast optical waveforms," IEEE J. Quantum Electron. 32, 161-172 (1996).
    [CrossRef]
  11. J. Paye and A. Migus, "Space-time Wigner functions and their application to the analysis of a pulse shaper," J. Opt. Soc. Am. B 12, 1480-1491 (1995).
    [CrossRef]
  12. R. D. Nelson, D. E. Leaird, and A. M. Weiner, "Programmable polarization-independent spectral phase compensation and pulse shaping," Opt. Express 11, 1763-1769 (2003).
    [CrossRef] [PubMed]
  13. H. P. Sardesai, C-.C Chang, and A. M. Weiner, "A femtosecond code-division multiple-access communication system testbed," J. Lightwave Technol. 16, 1953-1964 (1998).
    [CrossRef]
  14. Z. Bor and B. Racz, Opt. Commun. 54,165-170 (1985)
    [CrossRef]
  15. F.J. Harris, "Spectral analysis windowing," Wiley Encyclopedia of Electrical and Electronics Engineering, (Wiley, New York, 1999) Vol. 20, pp. 88-105.
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968)
  17. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, New Jersey, 1984)

2007 (2)

S. A. Diddams, L. Hollberg, and V. Mbele, "Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb," Nature 445, 627-630 (2007).
[CrossRef] [PubMed]

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, "Optical arbitrary waveform processing of more than 100 spectral comb lines," Nat. Photonics 1, 463-467 (2007).
[CrossRef]

2005 (2)

2003 (1)

2000 (2)

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

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis," Science 288,635-639 (2000).
[CrossRef] [PubMed]

1998 (1)

1996 (1)

M. M. Wefers and K. A. Nelson, "Space-time profiles of shaped ultrafast optical waveforms," IEEE J. Quantum Electron. 32, 161-172 (1996).
[CrossRef]

1995 (1)

1986 (2)

O. E. Martinez, "Grating and prism compressors in the case of finite beam size," J. Opt. Soc. Am. B 3, 929-934 (1986).
[CrossRef]

R. N. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, "Analysis of picoseconds pulse shape synthesis by spectral masking in a grating pulse compressor," IEEE J. Quantum Electron. 22, 682-696 (1986).
[CrossRef]

1985 (1)

Z. Bor and B. Racz, Opt. Commun. 54,165-170 (1985)
[CrossRef]

IEEE J. Quantum Electron. (2)

R. N. Thurston, J. P. Heritage, A. M. Weiner, and W. J. Tomlinson, "Analysis of picoseconds pulse shape synthesis by spectral masking in a grating pulse compressor," IEEE J. Quantum Electron. 22, 682-696 (1986).
[CrossRef]

M. M. Wefers and K. A. Nelson, "Space-time profiles of shaped ultrafast optical waveforms," IEEE J. Quantum Electron. 32, 161-172 (1996).
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. B (2)

Nat. Photonics (1)

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, "Optical arbitrary waveform processing of more than 100 spectral comb lines," Nat. Photonics 1, 463-467 (2007).
[CrossRef]

Nature (1)

S. A. Diddams, L. Hollberg, and V. Mbele, "Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb," Nature 445, 627-630 (2007).
[CrossRef] [PubMed]

Opt. Commun. (1)

Z. Bor and B. Racz, Opt. Commun. 54,165-170 (1985)
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Rev. Sci. Instrum. (1)

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

Science (1)

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis," Science 288,635-639 (2000).
[CrossRef] [PubMed]

Other (4)

J. Ye and S. T. Cundiff, eds., Femtosecond Optical Frequency Combs Technology: Principle, Operation, and Application (Springer, New York, 2004).

F.J. Harris, "Spectral analysis windowing," Wiley Encyclopedia of Electrical and Electronics Engineering, (Wiley, New York, 1999) Vol. 20, pp. 88-105.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968)

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, New Jersey, 1984)

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Figures (7)

Fig. 1.
Fig. 1.

Input spectrum and pulse train.

Fig. 2.
Fig. 2.

Dynamic masks and effective spectral filter functions. Left: the dynamic mask illustrates the abrupt change in the mask at time 0. Two cases are considered. For spectral amplitude masking, the mask is set to block every other comb line for t>0; the blue regions in the figure correspond to a mask value of 0. For spectral phase masking, the mask is set to impart a phase shift of π to every second comb line as illustrated by the blue regions in the figure. Right: the static filter functions corresponding to times t>0 illustrate the blurring of the effective mask for larger spot sizes. The effective spatial masks are calculated by convolving the smearing function in Eq. (11) with the spatial mask.

Fig. 3.
Fig. 3.

Response of the pulse train to an alternating amplitude mask, turned on abruptly at t=0, for various spot sizes, w 0. The dashed blue line shows the static pulse train where the full spectrum is allowed to pass, which yields the expected single pulsing. The dotted red line shows the static pulse train when every other comb line in the spectrum is masked out. This results in double pulsing behavior, with waveform fidelity that depends on w 0. The solid black line shows the dynamic response of a pulse train to the mask that abruptly switches at t=0.

Fig. 4.
Fig. 4.

Response of the pulse train to an alternating phase mask, turned on abruptly at t=0, at various spot sizes, w 0. The dashed blue line shows the static pulse train where the full spectrum is allowed to pass with no phase shift. The dotted red line shows the static pulse train when every other comb line in the spectrum is phase shifted by π; this yields the expected shift of half the period in the output pulse train. The solid black line shows the dynamic response of a pulse train to a mask that abruptly switches between the two at t=0.

Fig. 5.
Fig. 5.

Spectrogram response of a pulse train with equal size comb lines to a sliding spectral window of size frep for various spot sizes, w 0. The ideal case is a pseudo-spectrogram of what one would naively expect from a moving spectral filter.

Fig. 6.
Fig. 6.

Sliding filter effective masks at various w 0 as it the window crosses the center of the spectrum.

Fig. 7.
Fig. 7.

Electric field magnitude for a pulse train passed through a pulse shaper with a sliding spectral window of size frep for various spot sizes, w 0. Since the tunable filter ideally allows only one comb through at a time, the ideal pulse train would be converted to a constant magnitude, with no oscillation. We see that w 0=1/3 wrep is the closest we get to this ideal response with minimal oscillations in the region where the tunable filter shifts between comb lines.

Equations (15)

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e in ( x , t ) = R e { e ^ in ( x , t ) e j ω o t } = R e { a in ( t ) s ( x ) e j ω o t }
s ( x ) = e x 2 w in 2
e ̂ 1 ( x , t ) ~ d ω ˜ 2 π A in ( ω ˜ ) e ( x α ω ˜ ) 2 w 0 2 e j ω ˜ t
w 0 = ( f λ π w in ) ( cos θ i cos θ D )
α = λ 2 f 2 π cd cos θ D
e ̂ 2 ( x , t ) = M ( x ) e ̂ 1 ( x , t ) .
e ̂ out ( x , t ) = dx e ̂ 3 ( x , t ) u F ( x ) dxu F ( x ) u F ( x ) u F ( x ) .
A out ( ω ˜ ) = { 2 π w o 2 dx M ( x ) [ exp ( ( x α ω ˜ ) 2 w o 2 ) ] 2 } A in ( ω ˜ ) .
M ˜ ( x , ω ) = dt M ( x , t ) e j ω t .
A 2 ( x , ω ˜ ) ~ d ω ˜ 2 π A in ( ω ˜ ω ˜ ) M ( x , ω ˜ ) e ( x α ( ω ˜ ω ˜ ) ) 2 w o 2 .
A out ( ω ˜ ) ~ d ω ˜ 2 π dx A in ( ω ˜ ω ˜ ) M ( x , ω ˜ ) e ( x α ( ω ˜ ω ˜ ) ) 2 w o 2 e ( x α ω ˜ ) 2 w o 2 .
A out ( ω ˜ ) ~ d ω ˜ 2 π ( M ( ω ˜ ) e α 2 ( ω ˜ ) 2 2 w o 2 ) A in ( ω ˜ ω ˜ )
w rep = 2 π α f rep .
e in ( x , t ) = β a Re { d ω ˜ 2 π A ( ω ˜ ) s ( β a x ) e j γ ω ˜ x e j ( ω o + ω ˜ ) t }
S out ( x ) = j λ f d x s in ( x ) e jkxx f S in ( kx f )

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