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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 22 — Oct. 25, 2010
  • pp: 22988–22995
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Demonstration of high-fidelity dynamic optical arbitrary waveform generation

Nicolas K. Fontaine, David J. Geisler, Ryan P. Scott, Tingting He, Jonathan P. Heritage, and S. J. B. Yoo  »View Author Affiliations


Optics Express, Vol. 18, Issue 22, pp. 22988-22995 (2010)
http://dx.doi.org/10.1364/OE.18.022988


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Abstract

We experimentally demonstrate a dynamic line-by-line optical arbitrary waveform generation technique capable of generating continuous and bandwidth scalable high-fidelity waveforms without update rate limitations. Two quadrature modulators are used to create up to three spectral slices that are coherently combined by a passband-shaped multiplexer into a single contiguous spectrum to form complex optical waveforms with up to 30 GHz of bandwidth and 6 ns record lengths.

© 2010 OSA

1. Introduction

The generation of arbitrary optical waveforms with both scalable bandwidth (>1 THz) and long record lengths (> 1 ns) is an ongoing technical challenge with many new and exciting applications in telecommunications, optical signal processing, and microwave photonics [1

1. D. J. Geisler, N. K. Fontaine, T. He, R. P. Scott, L. Paraschis, J. P. Heritage, and S. J. B. Yoo, “Modulation-format agile, reconfigurable Tb/s transmitter based optical arbitrary waveform generation,” Opt. Express 17(18), 15911–15925 (2009). [CrossRef] [PubMed]

,2

2. P. J. Delfyett, S. Gee, Myoung-Taek Choi, H. Izadpanah, L. Wangkuen, S. Ozharar, F. Quinlan, and T. Yilmaz, “Optical frequency combs from semiconductor lasers and applications in ultrawideband signal processing and communications,” J. Lightwave Technol. 24(7), 2701–2719 (2006). [CrossRef]

]. Fourier domain pulse shaping techniques such as static optical arbitrary waveform generation (OAWG), also known as line-by-line pulse shaping, adjust the phase and intensity of each line in an optical frequency comb (OFC) using a Fourier domain waveform shaper (see Fig. 1(a)
Fig. 1 (a) Dynamic waveform shaper and dynamic-OAWG concept. (b) The spectral slice algorithm used to generate spectral slices. (c) Pre-emphasis of the spectral slices for the multiplexer and generation of temporal modulations. (d) Quadrature and (e) polar modulator.
for an example) to produce bandwidth scalable waveforms [1

1. D. J. Geisler, N. K. Fontaine, T. He, R. P. Scott, L. Paraschis, J. P. Heritage, and S. J. B. Yoo, “Modulation-format agile, reconfigurable Tb/s transmitter based optical arbitrary waveform generation,” Opt. Express 17(18), 15911–15925 (2009). [CrossRef] [PubMed]

,3

3. K. Takiguchi, K. Okamoto, T. Kominato, H. Takahashi, and T. Shibata, “Flexible pulse waveform generation using silica-waveguide-based spectrum synthesis circuit,” Electron. Lett. 40(9), 537–538 (2004). [CrossRef]

,4

4. Z. Jiang, C. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007). [CrossRef]

]. To output a stable waveform, the waveform shapers must maintain coherence between the comb lines. Operating with static modulations, the waveform shaper essentially acts as a highly reconfigurable filter which operates on the OFC. Therefore, the output is a repetitive waveform (i.e., shaped OFC) with a maximum waveform period equal to the input OFC’s period. Temporal multiplexing schemes which combine waveforms from the outputs of multiple waveform shapers [5

5. R. P. Scott, N. K. Fontaine, C. Yang, D. J. Geisler, K. Okamoto, J. P. Heritage, and S. J. B. Yoo, “Rapid updating of optical arbitrary waveforms via time-domain multiplexing,” Opt. Lett. 33(10), 1068–1070 (2008). [CrossRef] [PubMed]

,6

6. C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Time-multiplexed photonically enabled radio-frequency arbitrary waveform generation with 100 ps transitions,” Opt. Lett. 32(22), 3242–3244 (2007). [CrossRef] [PubMed]

], or high-resolution static waveform shapers [7

7. V. R. Supradeepa, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Femtosecond pulse shaping in two dimensions: towards higher complexity optical waveforms,” Opt. Express 16(16), 11878–11887 (2008). [CrossRef] [PubMed]

] have been used to demonstrate incremental increases in the record length of static-OAWG waveforms. However, because of their complexity and size, these techniques are impractical to scale to long record lengths and a different approach is needed.

2. The dynamic-OAWG concept

Figure 1 illustrates the dynamic-OAWG waveform shaper and the spectral-slice algorithm. A high-fidelity and practical dynamic waveform shaper must have a bandwidth-efficient modulator structure sufficiently fast to generate any spectral slice and a multiplexer with gapless spectral transmission. The waveform shaper modulates each line of an OFC to create spectral slices that, when coherently combined, will fill the gaps in the OFC’s discrete spectrum. Time-frequency duality ensures that if the target waveform’s spectrum is generated exactly, then the generated temporal waveform, d(t), is a perfect match to the target, a(t). Figure 1(a) shows the dynamic waveform shaper, which contains the spectral multiplexers and modulators that generate and coherently combine the different spectral slices. A demultiplexer with high adjacent channel isolation directs each comb line to a different channel. Then an array of modulators applies parallel temporal modulations to broaden each comb line into shaped spectral slices, and finally a multiplexer with gapless-transmission combines each slice. For this demonstration, we use in-phase (I) and quadrature-phase (Q) modulators and an arrayed-waveguide grating (AWG) with broadened passbands.

Figures 1(d), 1(e) show how to apply mn(t) to a comb line using the indicated drive signals for the quadrature (i.e., I/Q) modulator and polar (i.e., amplitude and phase) modulator. In an I/Q modulator (Fig. 1(d)), the two electrical drive signals that modulate the I and Q components of an optical carrier are simply equal to the real and imaginary parts of mn(t). They produce a spectral slice centered on a comb line, and it has twice the optical bandwidth as the electrical modulation bandwidth [16

16. R. G. Lyons, Understanding Digital Signal Processing, 2nd ed. (Prentice Hall PTR, Upper Saddle River, 2004).

] (i.e., positive and negative with respect to the carrier). In contrast, Fig. 1(e) shows a polar modulator, which is less efficient for producing any arbitrary spectral slice because it maps electrical signals to the magnitude and the phase of an optical carrier. In most cases, the driving signals are more complicated and not bandlimited. For example, frequency shifting a comb line requires endless, or modulo-2π, phase modulation.

3. Two-line dynamic-OAWG experiment

Figure 2(d) shows the heterodyne measurement technique used to simultaneously measure the full field (i.e., amplitude and phase) of the three separate outputs, A, B, and C [17

17. I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 013902 (2008). [CrossRef] [PubMed]

]. The average amplitude and phase of a repetitive signal, s(t), is measured against a reference line, r(t) = exp(j2πfRt), with approximately 32 GHz of optical measurement bandwidth. Here, s(t) is periodic with a 60 MHz repetition rate. The reference line is generated by frequency shifting the single-frequency laser by 35 MHz using an acousto-optic modulator. The balanced coherent receiver downconverts the signal comb to baseband and the detected photocurrent is digitized at 50 GS/s with 16 GHz of electrical bandwidth for 4 µs. The electrical spectrum (Fig. 2(d), bottom right) contains two sets of spectral comb lines with 60 MHz spacing: one set has frequencies greater than fR with a 25 MHz offset from dc, and the other set has frequencies less than fR with a 35 MHz offset from dc. The amplitude and phase of each comb line of S(ω) is selected from the electrical spectrum to recover the optical spectrum. Picking the peak value of each comb line effectively averages 240 of the 16.67-ns waveforms together (i.e., 4 µs averaging time). This heterodyne technique provides 32-GHz bandwidth, full-field measurements at a 15-Hz update rate (limited by computer processing time). Other techniques, such as optical arbitrary waveform measurement (OAWM) [18

18. N. K. Fontaine, R. P. Scott, L. Zhou, F. Soares, J. P. Heritage, and S. J. B. Yoo, “Real-time full-field arbitrary optical waveform measurement,” Nat. Photonics 4(4), 248–254 (2010). [CrossRef]

], are necessary for characterization of terahertz-bandwidth waveforms in real-time. Note that the fiber-pigtailed components between the deinterleaver and the AWG are susceptible to environmental fluctuations that cause a slowly varying phase drift between the two spectral slices which are not present in integrated devices. Therefore, we only show measurements with the correct relative phase which constitute approximately 8% of the measurements in the two-line experiment, and 1% of the measurements in the three-line experiment.

Figure 3
Fig. 3 Measurement results for the two line dynamic-OAWG experiments. (a) The entire 16 ns time record. Boxed area indicates the specified arbitrary waveform. (b,c) Isolation of the transform limited pulse with pre-emphasis (solid lines) and without pre-emphasis (dashed lines) for multiplexer filtering, Hn(ω). (d) Measured modulations, mn(t) at points (A) and (B) in Fig. 2(a). Comparison of the generated arbitrary waveform to the target waveform (grey) with (f) no iterations, and then (g,h) one iteration.
shows the measured results of a 20 GHz target waveform with a 6 ns record length consisting of a transform-limited pulse, a pulse with cubic spectral phase, and a pulse with quadratic spectral phase each with a 20th-order super-Gaussian spectral amplitude. The waveform has no resemblance to the input OFC; the waveform’s spectrum is continuous, without dips, and the temporal domain waveform periodicity is unrelated to the OFC period. Figure 3(a) shows the full time record (16.67 ns) which contains the fully specified arbitrary waveform in the first 8 ns. Due to the time multiplexing scheme used to provide the four signals to the two I/Q modulators, the second 8 ns contains another waveform that cannot be independently defined. Figures 3(b), 3(c) show the transform-limited pulse generated with and without applying pre-emphasis for the spectral multiplexer. With pre-emphasis for Hn(ω), the spectrum is flat and the time-domain ripple is reduced. Figure 3(d) shows the measured temporal modulations, mn(t) at points A and B that produce the two spectral slices shown in Fig. 3(e). The 1 GHz overlap of the modulated lines results from using 11-GHz wide spectral slice filters. Figure 3(f) shows that the measured waveform is a high-fidelity reproduction of the target waveform. Two figures of merit help quantify the fidelity of the generated waveforms, a normalized average energy error (NEE) [11

11. R. P. Scott, N. K. Fontaine, J. P. Heritage, and S. J. B. Yoo, “Dynamic optical arbitrary waveform generation and measurement,” Opt. Express 18(18), 18655–18670 (2010). [CrossRef] [PubMed]

] which represents a mean-squared error or total signal-to-noise ratio (SNR) and the normalized maximum energy error (MEE) that expresses the single largest deviation from the target waveform. The NEE is defined as the ratio between the error waveform’s [i.e., e(t) = d(t) − a(t)] energy and the energy in the target waveform, a(t), and the MEE is equal to the ratio between the peak energy of the error waveform and the peak energy of the target waveform. The waveform in Fig. 3(f) has an NEE less than 2% and a MEE of 4%. This level of accuracy is achieved using a thorough calibration of the waveform shaper components and it does not rely on any iterative algorithms or feedback. Waveform fidelity is increased by using feedback from this measurement to calculate a new set of modulations and Figs. 3(g), 3(h) shows the resulting waveform with NEE less than 0.6% and a MEE of 0.4%.

We choose to quantify fidelity by using metrics which analyze the energy in e(t) (i.e., NEE and MEE) because they include both phase and amplitude errors between a(t) and d(t) in a single number. Alternatively, phase and intensity errors of the generated waveforms could be calculated separately. For example, in Fig. 3(f) the maximum intensity error (MIE) between a(t) and d(t) (i.e., |d(t)|2 − |a(t)|2) compared to the peak intensity of a(t) is ~6% which is greater than the MEE of 0.4%. For other cases, when there is a mismatch in phase between a(t) and d(t) but very small intensity errors, the MEE is larger than the MIE. Additionally, phase and intensity metrics are more dependent on the waveform shape than NEE and MEE and are not as effective for comparing the fidelity of different waveforms.

Figure 4
Fig. 4 Additional dynamic-OAWG measurements. Target waveforms are grey. (a) Waveform with a −2.5 GHz/ns linear frequency chirp (20 GHz change over 8 ns). (b) Train of nine transform-limited pulses with a 300-ps spacing.
shows additional examples of dynamic-OAWG waveform generation to further emphasize that the shaped waveforms are arbitrary and do not resemble, or have any relation to, the periodicity to the input OFC. Figure 4(a) shows a waveform with quadratic spectral phase equivalent to a linear frequency chirp of −2.5 GHz/ns. The amount of quadratic spectral phase is large enough that each spectral component maps to a different temporal location. The generated waveform is a close match with the target (NEE = 0.7%, MEE = 4.7%) and the transition between the two spectral slices is nearly seamless. Figure 4(b) depicts a 300-ps spaced pulse train generated from an input OFC with a 105.5-ps period (NEE = 1.5%, MEE = 1.7%).

4. Three-line dynamic-OAWG experiment

Demonstrating true scalability to THz-bandwidth waveforms requires 100 I/Q modulators at a 10 GHz spacing, and a scheme to maintain the coherence between the slices either through integration, active stabilization, or a common-mode setup (i.e., a bulk-optics shaper). To demonstrate the bandwidth scalability of dynamic-OAWG, Fig. 5(a)
Fig. 5 (a) Three line dynamic-OAWG experiment. Dashed rectangles indicate the slices selected by the AWG. Shaped three-line waveform and comparison to target (grey) in the (b) temporal and (c) spectral domain.
shows a three-line experiment, which is an extension of the two-line experiment. The OFCG works by applying both strong amplitude and phase modulation to a single-frequency laser to generate a 10 GHz OFC with three lines of equal amplitude and approximately seven lines within −10 dB [19

19. T. Sakamoto, T. Kawanishi, and M. Izutsu, “Asymptotic formalism for ultraflat optical frequency comb generation using a Mach-Zehnder modulator,” Opt. Lett. 32(11), 1515–1517 (2007). [CrossRef] [PubMed]

]. The deinterleaver splits the even and odd comb lines to the upper and lower channels, respectively. A time-interleaving scheme similar to the one described previously, applies six independent modulations (corresponding to three sets of I and three sets of Q modulations) to generate three unique spectral slices. The upper I/Q modulator generates the center slice from the center even comb line, which is directed to input 2 of the AWG. The lower I/Q modulator generates slice 1 and slice 3, and a 1 × 2 splitter is used to direct them to input 1 and input 3 of the AWG multiplexer. At input 1 to the AWG, slice 1 is passed to the output and slice 3 is rejected with 30 dB of extinction [see AWG transmission in Fig. 2(c)]. Likewise, at input 3 of the AWG, slice 3 is passed and slice 1 is rejected.

Figures 5(b), 5(c) show high-fidelity generation of the target waveform. The waveform has approximately 30 GHz of optical bandwidth, 4 ns duration, and an NEE of 4% and a MEE of 4.7%. The greater error when compared to the two-line experiment is primarily attributed to the difficulty in aligning the constant spectral phase between the three slices. For example, the phase of slice 3 is slightly offset from the target phase [see the phase on the right side of Fig. 5(c)]. The waveform does not show degradation in fidelity due to the multiplexer filtering.

5. Conclusions

This experimental demonstration validates the dynamic-OAWG concepts and algorithms, which enable generation of continuous, high-fidelity waveforms in a bandwidth scalable fashion. Scaling to four spectral slices and beyond in a fiber-pigtailed configuration is difficult due to the complexities of coherently combining multiple spectral slices with a stable phase relationship. Implementing dynamic-OAWG in an integrated platform overcomes this issue; however, integration has many challenges including the development of an array of high-speed modulators, and individually accessing the modulators with low electrical crosstalk.

Dynamic-OAWG in conjunction with dynamic-OAWM [18

18. N. K. Fontaine, R. P. Scott, L. Zhou, F. Soares, J. P. Heritage, and S. J. B. Yoo, “Real-time full-field arbitrary optical waveform measurement,” Nat. Photonics 4(4), 248–254 (2010). [CrossRef]

], an analogous counterpart to OAWG in which a signal-spectrum is measured in slices against a reference comb, is a bandwidth scalable system for continuous generation and detection of arbitrary waveforms.

Acknowledgments

This work was supported in part by DARPA and SPAWAR under OAWG contract HR0011–05–C–0155.

References and links

1.

D. J. Geisler, N. K. Fontaine, T. He, R. P. Scott, L. Paraschis, J. P. Heritage, and S. J. B. Yoo, “Modulation-format agile, reconfigurable Tb/s transmitter based optical arbitrary waveform generation,” Opt. Express 17(18), 15911–15925 (2009). [CrossRef] [PubMed]

2.

P. J. Delfyett, S. Gee, Myoung-Taek Choi, H. Izadpanah, L. Wangkuen, S. Ozharar, F. Quinlan, and T. Yilmaz, “Optical frequency combs from semiconductor lasers and applications in ultrawideband signal processing and communications,” J. Lightwave Technol. 24(7), 2701–2719 (2006). [CrossRef]

3.

K. Takiguchi, K. Okamoto, T. Kominato, H. Takahashi, and T. Shibata, “Flexible pulse waveform generation using silica-waveguide-based spectrum synthesis circuit,” Electron. Lett. 40(9), 537–538 (2004). [CrossRef]

4.

Z. Jiang, C. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007). [CrossRef]

5.

R. P. Scott, N. K. Fontaine, C. Yang, D. J. Geisler, K. Okamoto, J. P. Heritage, and S. J. B. Yoo, “Rapid updating of optical arbitrary waveforms via time-domain multiplexing,” Opt. Lett. 33(10), 1068–1070 (2008). [CrossRef] [PubMed]

6.

C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Time-multiplexed photonically enabled radio-frequency arbitrary waveform generation with 100 ps transitions,” Opt. Lett. 32(22), 3242–3244 (2007). [CrossRef] [PubMed]

7.

V. R. Supradeepa, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Femtosecond pulse shaping in two dimensions: towards higher complexity optical waveforms,” Opt. Express 16(16), 11878–11887 (2008). [CrossRef] [PubMed]

8.

J. T. Willits, A. M. Weiner, and S. T. Cundiff, “Theory of rapid-update line-by-line pulse shaping,” Opt. Express 16(1), 315–327 (2008). [CrossRef] [PubMed]

9.

M. Akbulut, S. Bhooplapur, I. Ozdur, J. Davila-Rodriguez, and P. J. Delfyett, “Dynamic line-by-line pulse shaping with GHz update rate,” Opt. Express 18(17), 18284–18291 (2010). [CrossRef] [PubMed]

10.

N. K. Fontaine, “Optical arbitrary waveform generation and measurement,” Ph.D. dissertation (University of California, Davis (2010).

11.

R. P. Scott, N. K. Fontaine, J. P. Heritage, and S. J. B. Yoo, “Dynamic optical arbitrary waveform generation and measurement,” Opt. Express 18(18), 18655–18670 (2010). [CrossRef] [PubMed]

12.

F. M. Soares, J. H. Baek, N. K. Fontaine, X. Zhou, Y. Wang, R. P. Scott, J. P. Heritage, C. Junesand, S. Lourdudoss, K. Y. Liou, R. A. Hamm, W. Wang, B. Patel, S. Vatanapradit, L. A. Gruezke, W. T. Tsang, and S. J. B. Yoo, “Monolithically integrated InP wafer-scale 100-channel × 10-GHz AWG and Michelson interferometers for 1-THz-bandwidth optical arbitrary waveform generation,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2010).

13.

A. D. Ellis and F. C. G. Gunning, “Spectral density enhancement using coherent WDM,” IEEE Photon. Technol. Lett. 17(2), 504–506 (2005). [CrossRef]

14.

D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010). [CrossRef] [PubMed]

15.

X. Yi, N. K. Fontaine, R. P. Scott, and S. J. B. Yoo, “Tb/s coherent optical OFDM systems enabled by optical frequency combs,” J. Lightwave Technol. 28(14), 2054–2061 (2010). [CrossRef]

16.

R. G. Lyons, Understanding Digital Signal Processing, 2nd ed. (Prentice Hall PTR, Upper Saddle River, 2004).

17.

I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 013902 (2008). [CrossRef] [PubMed]

18.

N. K. Fontaine, R. P. Scott, L. Zhou, F. Soares, J. P. Heritage, and S. J. B. Yoo, “Real-time full-field arbitrary optical waveform measurement,” Nat. Photonics 4(4), 248–254 (2010). [CrossRef]

19.

T. Sakamoto, T. Kawanishi, and M. Izutsu, “Asymptotic formalism for ultraflat optical frequency comb generation using a Mach-Zehnder modulator,” Opt. Lett. 32(11), 1515–1517 (2007). [CrossRef] [PubMed]

OCIS Codes
(320.0320) Ultrafast optics : Ultrafast optics
(320.5540) Ultrafast optics : Pulse shaping

ToC Category:
Ultrafast Optics

History
Original Manuscript: July 26, 2010
Revised Manuscript: September 20, 2010
Manuscript Accepted: September 27, 2010
Published: October 15, 2010

Citation
Nicolas K. Fontaine, David J. Geisler, Ryan P. Scott, Tingting He, Jonathan P. Heritage, and S. J. B. Yoo, "Demonstration of high-fidelity dynamic 
optical arbitrary waveform generation," Opt. Express 18, 22988-22995 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-22988


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References

  1. D. J. Geisler, N. K. Fontaine, T. He, R. P. Scott, L. Paraschis, J. P. Heritage, and S. J. B. Yoo, “Modulation-format agile, reconfigurable Tb/s transmitter based optical arbitrary waveform generation,” Opt. Express 17(18), 15911–15925 (2009). [CrossRef] [PubMed]
  2. P. J. Delfyett, S. Gee, Myoung-Taek Choi, H. Izadpanah, L. Wangkuen, S. Ozharar, F. Quinlan, and T. Yilmaz, “Optical frequency combs from semiconductor lasers and applications in ultrawideband signal processing and communications,” J. Lightwave Technol. 24(7), 2701–2719 (2006). [CrossRef]
  3. K. Takiguchi, K. Okamoto, T. Kominato, H. Takahashi, and T. Shibata, “Flexible pulse waveform generation using silica-waveguide-based spectrum synthesis circuit,” Electron. Lett. 40(9), 537–538 (2004). [CrossRef]
  4. Z. Jiang, C. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007). [CrossRef]
  5. R. P. Scott, N. K. Fontaine, C. Yang, D. J. Geisler, K. Okamoto, J. P. Heritage, and S. J. B. Yoo, “Rapid updating of optical arbitrary waveforms via time-domain multiplexing,” Opt. Lett. 33(10), 1068–1070 (2008). [CrossRef] [PubMed]
  6. C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Time-multiplexed photonically enabled radio-frequency arbitrary waveform generation with 100 ps transitions,” Opt. Lett. 32(22), 3242–3244 (2007). [CrossRef] [PubMed]
  7. V. R. Supradeepa, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Femtosecond pulse shaping in two dimensions: towards higher complexity optical waveforms,” Opt. Express 16(16), 11878–11887 (2008). [CrossRef] [PubMed]
  8. J. T. Willits, A. M. Weiner, and S. T. Cundiff, “Theory of rapid-update line-by-line pulse shaping,” Opt. Express 16(1), 315–327 (2008). [CrossRef] [PubMed]
  9. M. Akbulut, S. Bhooplapur, I. Ozdur, J. Davila-Rodriguez, and P. J. Delfyett, “Dynamic line-by-line pulse shaping with GHz update rate,” Opt. Express 18(17), 18284–18291 (2010). [CrossRef] [PubMed]
  10. N. K. Fontaine, “Optical arbitrary waveform generation and measurement,” Ph.D. dissertation (University of California, Davis (2010).
  11. R. P. Scott, N. K. Fontaine, J. P. Heritage, and S. J. B. Yoo, “Dynamic optical arbitrary waveform generation and measurement,” Opt. Express 18(18), 18655–18670 (2010). [CrossRef] [PubMed]
  12. F. M. Soares, J. H. Baek, N. K. Fontaine, X. Zhou, Y. Wang, R. P. Scott, J. P. Heritage, C. Junesand, S. Lourdudoss, K. Y. Liou, R. A. Hamm, W. Wang, B. Patel, S. Vatanapradit, L. A. Gruezke, W. T. Tsang, and S. J. B. Yoo, “Monolithically integrated InP wafer-scale 100-channel × 10-GHz AWG and Michelson interferometers for 1-THz-bandwidth optical arbitrary waveform generation,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2010).
  13. A. D. Ellis and F. C. G. Gunning, “Spectral density enhancement using coherent WDM,” IEEE Photon. Technol. Lett. 17(2), 504–506 (2005). [CrossRef]
  14. D. Hillerkuss, M. Winter, M. Teschke, A. Marculescu, J. Li, G. Sigurdsson, K. Worms, S. Ben Ezra, N. Narkiss, W. Freude, and J. Leuthold, “Simple all-optical FFT scheme enabling Tbit/s real-time signal processing,” Opt. Express 18(9), 9324–9340 (2010). [CrossRef] [PubMed]
  15. X. Yi, N. K. Fontaine, R. P. Scott, and S. J. B. Yoo, “Tb/s coherent optical OFDM systems enabled by optical frequency combs,” J. Lightwave Technol. 28(14), 2054–2061 (2010). [CrossRef]
  16. R. G. Lyons, Understanding Digital Signal Processing, 2nd ed. (Prentice Hall PTR, Upper Saddle River, 2004).
  17. I. Coddington, W. C. Swann, and N. R. Newbury, “Coherent multiheterodyne spectroscopy using stabilized optical frequency combs,” Phys. Rev. Lett. 100(1), 013902 (2008). [CrossRef] [PubMed]
  18. N. K. Fontaine, R. P. Scott, L. Zhou, F. Soares, J. P. Heritage, and S. J. B. Yoo, “Real-time full-field arbitrary optical waveform measurement,” Nat. Photonics 4(4), 248–254 (2010). [CrossRef]
  19. T. Sakamoto, T. Kawanishi, and M. Izutsu, “Asymptotic formalism for ultraflat optical frequency comb generation using a Mach-Zehnder modulator,” Opt. Lett. 32(11), 1515–1517 (2007). [CrossRef] [PubMed]

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