## The impact of optical comb stability on waveforms generated via spectral line-by-line pulse shaping

Optics Express, Vol. 14, Issue 26, pp. 13164-13176 (2006)

http://dx.doi.org/10.1364/OE.14.013164

Acrobat PDF (884 KB)

### 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

## 1. Introduction and problem description

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]

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]

_{π}~5V at 10 GHz) driven at f

_{rep}=9 GHz (~1.6 V

_{π}peak to peak) to obtain an optical comb with lines spaced by 9 GHz as shown in Fig. 1(b). The frequencies are defined relative to the CW laser frequency. The comb spectrum is observed using an optical spectrum analyzer (OSA) with 0.01 nm resolution. Lines at frequencies higher than the CW laser are all in phase as measured using the method reported in Ref. [7

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]

_{rep}is fixed, we are able to emulate the effects of the frequency offsets in the mode-locked laser of [4

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]

_{rep}is the waveform periodicity and Φ is the phase applied to one line. τ(Φ) due to the phase settings (Φ=0, π) are shown with dashed lines

## 2. Modeling and fitting

### 2.1 Modeling

_{0}is the Gaussian beam radius (half-width at 1/e

^{2}of intensity) and α is the spatial dispersion with units cm (rad/s)

^{-1}[2

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

_{0}=75 µm. Each LCM pixel has a 100 µm width and a corresponding frequency span of 4.5 GHz; therefore frequency lines are centered on every second LCM pixel. Our line-by-line shaper has α=3.54e-13 cm (rad/s)

^{-1}and a full-width at half maximum spectral resolution of 5.31GHz. The spectral resolution is limited here by the width of turning on one single LCM pixel. This obtained resolution ensures observation of temporal overlapping contributions from adjacent pulses. The output fields of the linear filter E

_{out}(ω) is the product of the input field E

_{in}(ω) and the filter function H(ω):

_{out}(t) is obtained by Fourier transforming the output field:

_{0}(blue dash) are both depicted as arrows. The comb line positions can be described as [1

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]

_{rep}is the repetition frequency, m is a large integer, and ε is the absolute frequency comb offset. For our purpose, however, we note that the pulse shaper acts as a frequency filter (schematically shown in red solid), whose center frequency provides a frequency reference, which we denote as f

_{s}(red dash). To optimally select two comb lines via pulse shaping, the comb frequencies {f

_{m}, f

_{m+1}} are aligned symmetrically about f

_{s}by fine tuning the LCM position. This means that line frequencies referenced to f

_{s}are:

*f*̃≡f-f

_{s}. After alignment, if the comb offset varies from its initial value to ε′=ε+δε (green solid), what is observed by the pulse shaper is only the frequency offset change δε. Nonzero values of δε will eventually induce power loss as the lines approach the edges of the filter function, and this will transform into waveform variations in the time-domain. It is important to emphasize that only the relative frequency offset is important in line-by-line pulse shaping, not the absolute offset.

_{r}in the optical comb will effectively be transferred to an extra energy loss by the central frequency discriminator. This loss viewed in the frequency-domain will result in large peak intensity variation in the time domain.

### 2.2 Data fitting and discussions

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]

_{rep}. When one of the spectral lines has an applied π phase shift relative to the other line, the field of the filtered comb is depicted in Fig. 5(b), with one line having negative amplitude. The corresponding time-domain intensity is shown as the blue solid trace in Fig. 5(d), a cosine function having period of T with delay τ(Φ)=T/2. Figure 5(c) shows the resulting filtered comb with offset δε, constituting three lines denoted as {i, ii, iii}. Note that the amplitudes of all the lines are affected by the frequency-dependent transmission of the filter, while the signs read {+,+,-}, respectively. The corresponding time-domain intensity for the lines with this particular phase sequence, shown as the green dashed trace in Fig. 5(d), yields a shifted cosine function now with periodicity of T/2. This phenomenon can be understood from the beating between each two lines: the 9 GHz beating of lines {i, ii} almost cancels with that from {ii, iii}; what is left is the remaining 18 GHz beating between lines {i, iii}. The ability to predict subtle details of the experiment, such as the evolution of the intensity periodicity from T to T/2, again demonstrates the power of our frequency-domain treatment.

### 2.3 Correlating frequency and time-domain pictures: intermediate phases

^{2}(Φ/2) (red dashed line) in Fig. 6(b). Using the linear dependence of τ(Φ) on Φ, we infer that for an ideal comb, the range of intensity variation with frequency offset follows a temporal mask of the form cos

^{2}(πf

_{rep}τ). We will use this empirical observation as an aid to interpret some of our later results.

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]

## 3. Time-domain waveforms designed for monitoring optical frequency fluctuations

### 3.1 Design approach

_{rep}, where N is an integer. For the N=1 example discussed above, we have shown that there is significant sensitivity to frequency fluctuations at time positions equal to odd multiples of T/2 but not at multiples of T. Let us assume that this behavior applies also for N>1, which is justified by the results given below. Then any two lines spaced apart by 2f

_{rep}or greater will exhibit waveform positions both sensitive and insensitive to comb fluctuations, since there will be peaks situated at multiples of T/N. Furthermore, applying a phase shift onto one of the two lines allows us to finely shift the positions of the peaks for optimum sensitivity.

_{rep}=9 GHz spacing (T=111.1 ps) is assumed with spectral comb lines of equal amplitudes. Figure 7(a) shows a filter designed to select two lines spaced by 2f

_{rep}(N=2) which removes the intermediate line. Optical frequency comb offsets of {0–50}% with 10% per step are indicated. The corresponding time-domain waveforms are shown in Fig. 7(b). I(0) is invariant while I(T/2) is sensitive to frequency fluctuations. The cos

^{2}(πf

_{rep}t) is plotted (green dashed line) to emphasize the temporal intensity variation relation discussed in the previous section. The ratio of I(T/2) to I(0) offers a clear indication of frequency offsets, especially for large offsets. Figure 7(c) shows the semi-log plot of the contrast ratio with various offset values. The scenario is different if one of the lines is π-shifted, shown in Fig. 7(d). Since the waveforms are delayed by T/4, all the peaks encounter the same amount of intensity variation. One interesting observation is how the cos

^{2}(πf

_{rep}t) function nicely cuts the intensity waveform at 50% offset. This once again supports our conclusion drawn from the previous section.

### 3.2 Experimental results

_{π}lithium niobate phase modulator (20GHz bandwidth, V

_{π}of ~2.8V at 1 GHz) is modulated at f

_{rep}=9GHz with a driving voltage of ~2.6 V

_{π}peak to peak to obtain the spectrum shown in Fig. 8(a). Relative phases of the lines are measured for lines {-6 to 5}. Lines {1, 0, -3, and -5} are π out of phase with the rest, which agree with calculation using a Bessel series [8

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]

_{rep}.

### 3.3 Larger N values

^{2}(πf

_{rep}t) function. The time-domain picture is clearly illustrated: frequency sensitivity of waveforms generated via line-by-line pulse shaping in the 100% duty cycle regime is least noticeable at time positions equal to 0 (or integral multiple of the repetition period T) and increasingly noticeable for increasing time offsets. Maximum intensity variations with frequency offsets occur for time positions of T/2 and its odd multiples.

## 4. Conclusion

## Acknowledgments

## References and links

1. | S. T. Cundiff, J. Ye, and J. L. Hall, “Optical frequency synthesis based on mode-locked lasers,” Rev. Sci. Inst. |

2. | A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Inst. |

3. | Z. Jiang, D. E. Leaird, and A. M. Weiner, “Line-by-line shaping control for optical arbitrary waveform generation,” Opt. Express |

4. | Z. Jiang, D. S. Seo, D. E. Leaird, and A. M. Weiner, “Spectral line-by-line pulse shaping,” Opt. Lett. |

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. |

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. |

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. |

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. |

**OCIS Codes**

(070.4790) Fourier optics and signal processing : Spectrum analysis

(120.3930) Instrumentation, measurement, and metrology : Metrological instrumentation

(140.4050) Lasers and laser optics : Mode-locked lasers

(230.2090) Optical devices : Electro-optical devices

(320.5540) Ultrafast optics : Pulse shaping

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: October 13, 2006

Manuscript Accepted: December 10, 2006

Published: December 22, 2006

**Citation**

Chen-Bin Huang, Zhi Jiang, Daniel E. Leaird, and Andrew M. Weiner, "The impact of optical comb stability on
waveforms generated via spectral line-by-line
pulse shaping," Opt. Express **14**, 13164-13176 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-26-13164

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### References

- 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]
- 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]
- 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]
- 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]
- 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]

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