Abstract

Optical arbitrary waveform generation using the line-by-line pulse shaping technique has been shown to be sensitive to variations in the offset frequency of the input frequency comb due to time-domain waveform interference. Here we present a frequency-domain model that is able to predict waveform changes arising from offset frequency variations. In experiments we controllably shift the frequency of a comb derived from a phase-modulated CW laser, which allows us to quantitatively investigate waveforms generated by pulse shaping as a function of offset frequency. Experimental data are in excellent agreement with the predictions of our frequency-domain model. In addition, we propose and analyze new waveforms designed for monitoring of offset frequency variations by pulse shaping.

© 2006 Optical Society of America

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References

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  1. S. T. Cundiff, J. Ye, and J. L. Hall, "Optical frequency synthesis based on mode-locked lasers," Rev. Sci. Inst. 72, 3749-3771 (2001).
    [CrossRef]
  2. A. M. Weiner, "Femtosecond pulse shaping using spatial light modulators," Rev. Sci. Inst. 71, 1929-1960 (2000).
    [CrossRef]
  3. Z. Jiang, D. E. Leaird, and A. M. Weiner, "Line-by-line shaping control for optical arbitrary waveform generation," Opt. Express 13, 10431-10439 (2005).
    [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. S. Hisatake, Y. Nakase, K. Shibuya, and T. Kobayashi, "Generation of flat power-envelope terahertz-wide modulation sidebands from a continuous-wave laser based on an external electro-optic phase modulator," Opt. Lett. 30, 777-779 (2005).
    [CrossRef] [PubMed]
  6. Z. Jiang, D. E. Leaird, and A. M. Weiner, "Optical processing based on spectral line-by-line pulse shaping on a phase modulated CW laser," IEEE J. Quantum Electron. 42, 657-665 (2006).
    [CrossRef]
  7. Z. Jiang, D. E. Leaird, and A. M. Weiner, "Optical arbitrary waveform generation and characterization using spectral line-by-line control," IEEE J. Lightwave Technol. 24, 2487-2494 (2006).
    [CrossRef]
  8. T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, "Optical pulse compression using high-frequency electrooptic phase modulation," IEEE J. Quantum Electron. 24, 382-387 (1988).
    [CrossRef]

2006

Z. Jiang, D. E. Leaird, and A. M. Weiner, "Optical processing based on spectral line-by-line pulse shaping on a phase modulated CW laser," IEEE J. Quantum Electron. 42, 657-665 (2006).
[CrossRef]

Z. Jiang, D. E. Leaird, and A. M. Weiner, "Optical arbitrary waveform generation and characterization using spectral line-by-line control," IEEE J. Lightwave Technol. 24, 2487-2494 (2006).
[CrossRef]

2005

2001

S. T. Cundiff, J. Ye, and J. L. Hall, "Optical frequency synthesis based on mode-locked lasers," Rev. Sci. Inst. 72, 3749-3771 (2001).
[CrossRef]

2000

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

1988

T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, "Optical pulse compression using high-frequency electrooptic phase modulation," IEEE J. Quantum Electron. 24, 382-387 (1988).
[CrossRef]

Amano, K.

T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, "Optical pulse compression using high-frequency electrooptic phase modulation," IEEE J. Quantum Electron. 24, 382-387 (1988).
[CrossRef]

Cundiff, S. T.

S. T. Cundiff, J. Ye, and J. L. Hall, "Optical frequency synthesis based on mode-locked lasers," Rev. Sci. Inst. 72, 3749-3771 (2001).
[CrossRef]

Fukushima, Y.

T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, "Optical pulse compression using high-frequency electrooptic phase modulation," IEEE J. Quantum Electron. 24, 382-387 (1988).
[CrossRef]

Hall, J. L.

S. T. Cundiff, J. Ye, and J. L. Hall, "Optical frequency synthesis based on mode-locked lasers," Rev. Sci. Inst. 72, 3749-3771 (2001).
[CrossRef]

Hisatake, S.

Jiang, Z.

Z. Jiang, D. E. Leaird, and A. M. Weiner, "Optical processing based on spectral line-by-line pulse shaping on a phase modulated CW laser," IEEE J. Quantum Electron. 42, 657-665 (2006).
[CrossRef]

Z. Jiang, D. E. Leaird, and A. M. Weiner, "Optical arbitrary waveform generation and characterization using spectral line-by-line control," IEEE J. Lightwave Technol. 24, 2487-2494 (2006).
[CrossRef]

Z. Jiang, D. E. Leaird, and A. M. Weiner, "Line-by-line shaping control for optical arbitrary waveform generation," Opt. Express 13, 10431-10439 (2005).
[CrossRef] [PubMed]

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]

Kobayashi, T.

S. Hisatake, Y. Nakase, K. Shibuya, and T. Kobayashi, "Generation of flat power-envelope terahertz-wide modulation sidebands from a continuous-wave laser based on an external electro-optic phase modulator," Opt. Lett. 30, 777-779 (2005).
[CrossRef] [PubMed]

T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, "Optical pulse compression using high-frequency electrooptic phase modulation," IEEE J. Quantum Electron. 24, 382-387 (1988).
[CrossRef]

Leaird, D. E.

Z. Jiang, D. E. Leaird, and A. M. Weiner, "Optical arbitrary waveform generation and characterization using spectral line-by-line control," IEEE J. Lightwave Technol. 24, 2487-2494 (2006).
[CrossRef]

Z. Jiang, D. E. Leaird, and A. M. Weiner, "Optical processing based on spectral line-by-line pulse shaping on a phase modulated CW laser," IEEE J. Quantum Electron. 42, 657-665 (2006).
[CrossRef]

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]

Z. Jiang, D. E. Leaird, and A. M. Weiner, "Line-by-line shaping control for optical arbitrary waveform generation," Opt. Express 13, 10431-10439 (2005).
[CrossRef] [PubMed]

Morimoto, A.

T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, "Optical pulse compression using high-frequency electrooptic phase modulation," IEEE J. Quantum Electron. 24, 382-387 (1988).
[CrossRef]

Nakase, Y.

Seo, D. S.

Shibuya, K.

Sueta, T.

T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, "Optical pulse compression using high-frequency electrooptic phase modulation," IEEE J. Quantum Electron. 24, 382-387 (1988).
[CrossRef]

Weiner, A. M.

Z. Jiang, D. E. Leaird, and A. M. Weiner, "Optical processing based on spectral line-by-line pulse shaping on a phase modulated CW laser," IEEE J. Quantum Electron. 42, 657-665 (2006).
[CrossRef]

Z. Jiang, D. E. Leaird, and A. M. Weiner, "Optical arbitrary waveform generation and characterization using spectral line-by-line control," IEEE J. Lightwave Technol. 24, 2487-2494 (2006).
[CrossRef]

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]

Z. Jiang, D. E. Leaird, and A. M. Weiner, "Line-by-line shaping control for optical arbitrary waveform generation," Opt. Express 13, 10431-10439 (2005).
[CrossRef] [PubMed]

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

Yao, H.

T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, "Optical pulse compression using high-frequency electrooptic phase modulation," IEEE J. Quantum Electron. 24, 382-387 (1988).
[CrossRef]

Ye, J.

S. T. Cundiff, J. Ye, and J. L. Hall, "Optical frequency synthesis based on mode-locked lasers," Rev. Sci. Inst. 72, 3749-3771 (2001).
[CrossRef]

IEEE J. Lightwave Technol.

Z. Jiang, D. E. Leaird, and A. M. Weiner, "Optical arbitrary waveform generation and characterization using spectral line-by-line control," IEEE J. Lightwave Technol. 24, 2487-2494 (2006).
[CrossRef]

IEEE J. Quantum Electron.

T. Kobayashi, H. Yao, K. Amano, Y. Fukushima, A. Morimoto, and T. Sueta, "Optical pulse compression using high-frequency electrooptic phase modulation," IEEE J. Quantum Electron. 24, 382-387 (1988).
[CrossRef]

Z. Jiang, D. E. Leaird, and A. M. Weiner, "Optical processing based on spectral line-by-line pulse shaping on a phase modulated CW laser," IEEE J. Quantum Electron. 42, 657-665 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Rev. Sci. Inst.

S. T. Cundiff, J. Ye, and J. L. Hall, "Optical frequency synthesis based on mode-locked lasers," Rev. Sci. Inst. 72, 3749-3771 (2001).
[CrossRef]

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

Supplementary Material (4)

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Figures (10)

Fig. 1.
Fig. 1.

(a). Schematic of experimental setup. PM: phase modulator; PS: reflective line-by-line pulse shaper; DET: to OSA or sampling scope. (b). Optical spectrum of the phase-modulated CW laser frequency scale relative to the CW line. The frequency lines are separated by 9GHz. Inset: lines circled are expanded and plotted on a linear scale. Lines {1, 2} are selected by the line-by-line shaper.

Fig. 2.
Fig. 2.

Experimental results by detuning the PMCW comb with relative frequency offsets of {0, 7, 14, 21, 28, 35, 42 and 49}%. (a, c) Spectra (linear scale in relative frequency) for Φ=0 and Φ=π, respectively. The horizontal arrows indicate the initial line positions and the direction of detuning. (b, d) Time-domain intensities for corresponding frequency offsets for Φ=0 and Φ=π, respectively. The vertical arrows indicate intensity peak variations as the optical frequency offset increases. Zero delay positions are marked by dashed lines. (Video file of 0.96 Mb for Φ=0 and 0.98 Mb for Φ=π.)

Fig. 3.
Fig. 3.

(a). Optical comb in absolute frequency position (black solid arrows) and having an offset e (blue dashed arrows). The schematic filter function is shown in red. (b) Two spectral lines selected by the pulse shaper are plotted against relative frequency after aligning the LCM (no additional offset: blue dash; with additional offset of δε: green solid). Mask function (solid line) and Gaussian intensity profile with radius w0=75um (dashed line) for Φ=0 (c) for both lines and for Φ=π (e) to one line. Resulting 4-pixel passband filter function |H(ω)|2: (d) experimental (solid line) result by sweeping the wavelength of a CW laser and simulated (dashed line) for Φ=0; (f) experimental (solid line) and simulated (dashed line) for Φ=π. The arrows schematically indicate the comb line positions.

Fig. 4.
Fig. 4.

Fit to Fig. 2 using the frequency-domain model. (a, c) Spectra fit (linear scale) for δε of {0, 7, 14, 21, 28, 35, 42 and 49}%. The arrows indicate the initial line positions and the direction of detuning. Experimental time-domain traces for corresponding frequency offsets are shown for Φ=0 (b) and Φ=π (d). The arrows indicate intensity peak variations associated with these frequency offsets. Delay positions are marked by dashed lines. (Video file of 0.97 Mb for Φ=0 and 1.00 Mb for Φ=π.)

Fig. 5.
Fig. 5.

Schematic of the concept of periodicity conversion due to shifting of the optical frequency comb when a phase shift is applied to one comb line. (a) Initial and the shifted frequency comb with fixed line amplitudes depicted by 4 arrows (solid: initial; dashed: shifted). The filter function is shown in red dashed line. (b) Filtered comb without frequency shift. (c) Filtered comb with a large frequency shift. (d) Simulated time-domain results showing periodicity conversion: from initial comb without frequency shift (solid) and the shifted comb (dash).

Fig. 6.
Fig. 6.

(a). Experimental (symbols: Φ={0, π/4, π/2, 3π/4 and π}) and calculated (lines: Φ=0 to π in π/4 increments) waveform amplitudes at corresponding τ(Φ) for δε of {0, 7, 14, 21, 28, 35, 42 and 49}%. (b). Intensity values for 49% frequency offset at corresponding τ(Φ) for different Φ values using phase-modulated CW comb (blue circle: experimental; blue solid line: calculation). Calculated (dotted line) and experimental (diamonds) τ(Φ) is also included. Intensity values assuming ideal comb lines (red square) is plotted with cos2(Φ/2) (red dashed line).

Fig. 7.
Fig. 7.

Two lines selected at 2frep by blocking the center line. (a) Ideal comb lines shifted with δε {0–50}% in 10% increment with no phase control. (b) Resulting time-domain intensity: peak at time zero is stable while peak at time T/2 is sensitive to offsets. (c) Semi-log plot of I(T/2)/I(0). (d) Ideal comb lines shifted with offsets {0–50}% in 10% increment with Φ=π to one line. In (b, d), green dashed lines depict cos2(πfrept).

Fig. 8.
Fig. 8.

(a). Spectrum obtained from a low Vπ phase modulator. Circled spectral portion is expanded in (b), in linear scale and normalized to line {3}. Lines {2, 4} are selected by the line-by-line shaper while the others are suppressed.

Figs. 9.
Figs. 9.

(a). and (b). Experimental spectra and the lines used for calculation (linear scale) for two lines with spacing of 2frep with δε of {0, 7, 14, 21, 28, 35, 42, and 49}%. The horizontal arrows indicate the initial line positions and the direction of detuning. (c–e) Experimental time-domain waveforms for Φ={0, π/2, and π}, respectively. (f–h) Calculated time-domain waveforms for Φ={0, π/2, and π}, respectively. The vertical arrows indicate intensity variations as the optical frequency offset increases for different temporal positions. Temporal positions: dash-dot: zero delay; dashed: T/2.

Fig. 10.
Fig. 10.

Simulation spectra for two lines with N=3 (a) and N=4 (e) with δε of {0–50}% in 10% increments. Resulting time-domain waveforms are given for N=3 (b–d) and N=4 (f–h) with phase shifts applied to one line Φ={0, π/2, π}, respectively. Green dashed lines depict cos2(πfrept). Temporal positions: dash-dot: zero delay; dashed: T/2.

Equations (6)

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τ ( Φ ) = Φ 2 π T
H ( ω ) = ( 2 π w 0 2 ) 1 2 M ( x ) e 2 ( x α ω ) 2 w 0 2 d x
E out ( ω ) = E in ( ω ) H ( ω )
e out ( t ) = 1 2 π E out ( ω ) e j ω t d ω
f m = m f rep + ε
f ~ m = ( m 1 2 ) f rep

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