## Dynamic optical arbitrary waveform generation and measurement |

Optics Express, Vol. 18, Issue 18, pp. 18655-18670 (2010)

http://dx.doi.org/10.1364/OE.18.018655

Acrobat PDF (6613 KB)

### Abstract

We introduce a dynamic optical arbitrary waveform generation (OAWG) technique that produces bandwidth scalable, continuous waveforms of near perfect fidelity. Additionally, OAWG’s complement, real-time arbitrary optical waveform measurement (OAWM) is discussed. These approaches utilize gigahertz-bandwidth electronics to generate, or measure, truly arbitrary and dynamic optical waveforms scalable to terahertz bandwidths and infinite record lengths. We describe the theory, algorithms and enabling technologies necessary to calculate and produce a set of spectral modulations that create continuous, high-fidelity waveforms in the presence of spectral filtering from multiplexers.

© 2010 OSA

## 1. Introduction

1. 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]

7. 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]

*dynamic-OAWG*, a technique to achieve high-fidelity, continuous generation of arbitrary optical waveforms of scalable bandwidth using optical frequency combs, spectral multiplexers, and an array of modulators. We present theory and simulations to demonstrate that there are actually numerous sets of comb line modulations that generate the same waveform. By example, we show how our dynamic-OAWG algorithm produces optimum modulations for high-fidelity continuous waveforms, even in the presence of multiplexer filtering. Additionally, there are discussions of enabling technologies for dynamic-OAWG including electro-optical modulators and spectral multiplexers. The final section of the paper describes an analogous real-time waveform characterization technique called optical arbitrary waveform measurement (OAWM) [8

8. 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]

10. N. K. Fontaine, D. J. Geisler, R. P. Scott, T. He, J. P. Heritage, and S. J. B. Yoo, “Demonstration of high-fidelity dynamic optical arbitrary waveform generation,” Opt. Express (submitted for publication). [PubMed]

### 1.1. A brief review of static-OAWG

1. 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]

11. N. K. Fontaine, R. P. Scott, C. Yang, D. J. Geisler, J. P. Heritage, K. Okamoto, and S. J. B. Yoo, “Compact 10 GHz loopback arrayed-waveguide grating for high-fidelity optical arbitrary waveform generation,” Opt. Lett. **33**(15), 1714–1716 (2008). [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), Paper OThS1.

2. Z. Jiang, D. E. Leaird, and A. M. Weiner, “Line-by-line pulse shaping control for optical arbitrary waveform generation,” Opt. Express **13**(25), 10431–10439 (2005). [CrossRef] [PubMed]

6. 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**(8), 463–467 (2007). [CrossRef]

### 1.2. Dynamically updating static-OAWG

3. P. J. Delfyett, S. Gee, C. Myoung-Taek, 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]

13. T. He, N. K. Fontaine, R. P. Scott, D. J. Geisler, J. P. Heritage, and S. J. B. Yoo, “Optical arbitrary waveform generation based packet generation and all-optical separation for optical-label switching,” IEEE Photon. Technol. Lett. **22**(10), 715–717 (2010). [CrossRef]

15. 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]

16. 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]

*N*× 1 star coupler can be used at the output of the waveform shaper since a standard spectral multiplexer would filter the modulations, thereby distorting the waveform. (3) Subwaveforms have excessive bandwidth since the process of temporally slicing the long arbitrary waveform with bandwidth,

*B*, into subwaveforms introduces instantaneous transitions at the edges. Therefore, the static-OAWG representation of each subwaveform is not necessarily bandlimited to

*B*. Section 5.3 provides details and simulations that clearly demonstrate the three problems listed above. All of the disadvantages of rapid-update static-OAWG are avoided by approaching the dynamic generation of waveforms from a new, and perhaps broader, perspective that we call dynamic-OAWG. The following sections describe the concept in detail, including enabling technologies, algorithms for calculating the spectral modulations, and supporting simulations.

## 2. The dynamic-OAWG concept

*I/Q*signals of Fig. 2(d). A signal multiplexer,

*H*(

*ω*), coherently combines the modulated lines (now spectral slices) into a single contiguous complex spectrum that exactly matches the target waveform. To determine the required electrical

*I/Q*modulations, the target waveform is processed by a digital signal processor (DSP) in the following manner: the continuous target waveform [Fig. 2(b)] is temporally sliced into long subwaveforms (i.e., many OFC periods). The complex spectrum of each subwaveform is calculated [Fig. 2(c)] and subsequently spectrally sliced using a set of spectral slice filters. Here, the spectral slice filters are gapless and their center-to-center spacing is equal to the OFC spacing. The modulations necessary to create each complex spectral slice are pre-emphasized for the multiplexer [Fig. 2(d)] before a digital-to-analog converter (DAC) array produces each of the electrical signals. The output sample rate must be equivalent to, or greater than, the slice bandwidth (i.e., Nyquist rate for quadrature sampling [17]). The temporal modulations of adjacent subwaveforms are stitched together and when done correctly, the resulting continuous modulations maintain their fidelity.

*B*. In dynamic-OAWG,

*B*is equal to the number of spectral slices multiplied by the slice spacing. Thus, to achieve a desired waveform bandwidth or temporal resolution, there is a tradeoff between the number of spectral slices (more complex optics for more slices) and the slice bandwidth (more complex electronics for broader slice bandwidth) which must be optimized. The 10 GHz slice spacing indicated in Fig. 2 falls in the frequency range where optical demultiplexers, modulators, and DACs overlap, while simultaneously maximizing the generation bandwidth and minimizing the system complexity. For instance, the fastest commercial electronic DACs have 5–10 GHz of analog bandwidth and set an upper limit on the slice spacing. Higher resolution AWGs require larger areas and set the lower limit of the slice spacing. By studying the dynamic-OAWG algorithm it is possible to optimize OAWG device designs and determine the best tradeoffs between the various system parameters.

## 3. Enabling technologies for dynamic-OAWG

### 3.1. Optical modulation

*a*(

*t*), as a 400-ps-long trajectory of a vector which traverses all four quadratures within the real-imaginary plane. This trajectory could represent the modulation necessary to produce a single spectral slice such as that shown in Fig. 2(a). Figure 3(b) shows the waveform’s complex spectrum which has a total optical bandwidth of 10 GHz.

*I/Q*) modulator. It consists of nested push-pull Mach-Zehnder modulators (MZM) with their outputs combined 90° out of phase. One MZM manipulates the

*I*-field while the other MZM manipulates the

*Q*-field and since the

*I/Q*modulator uses direct mapping of the scalar signals to the optical field (when operating in its linear regime); it is a spectrally efficient structure. Quadrature modulation has the added advantage that the required electrical bandwidth per MZM necessary to recreate a particular trajectory is one-half of the total optical bandwidth of that trajectory [17]. Thus, we use a slightly more complicated structure to minimize the required electrical signal complexity and bandwidth. Figure 4(e) shows the

*I/Q*modulation signals required to reproduce the trajectory of Fig. 3(a). Figure 4(f) presents the calculated modulation signals’ electrical power spectra where both

*I*and

*Q*are each bandlimited to 5 GHz. Another, often overlooked, advantage of the

*I/Q*modulator is that it is possible to trade increased insertion loss for a reduced drive voltage requirement. When using

*I/Q*modulation, reaching all possible phase values does not require a peak phase modulation of ± π rad. Instead, you can restrict the maximum achievable amplitude, and still cover a full circle (centered on the origin) in the real-imaginary plane. Thus it is possible to reproduce an amplitude scaled version of any trajectory.

### 3.2. Spectral multiplexers

*f*pulse shaper configuration [2

2. Z. Jiang, D. E. Leaird, and A. M. Weiner, “Line-by-line pulse shaping control for optical arbitrary waveform generation,” Opt. Express **13**(25), 10431–10439 (2005). [CrossRef] [PubMed]

19. M. K. Smit and C. Van Dam, “PHASAR-based WDM-devices: Principles, design and applications,” IEEE J. Sel. Top. Quantum Electron. **2**(2), 236–250 (1996). [CrossRef]

20. K. Okamoto, “Recent progress of integrated optics planar lightwave circuits,” Opt. Quantum Electron. **31**(2), 107–129 (1999). [CrossRef]

21. N. K. Fontaine, Y. Jie, J. Wei, D. J. Geisler, K. Okamoto, H. Ray, and S. Yoo, “Active arrayed-waveguide grating with amplitude and phase control for arbitrary filter generation and high-order dispersion compensation,” in *34th European Conference on Optical Communication (ECOC 2008)*, Technical Digest (CD) (IEEE, 2008), 1–2. http://dx.doi.org/10.1109/ECOC.2008.4729152

## 4. Mathematical framework for dynamic-OAWG

*a*(

_{R}*t*) is defined aswhere

*R*is a coefficient that defines the amplitude and phase of the

_{n}*n*-th comb line and Δ

*ω*is the comb line spacing in the frequency domain. (i.e., Δ

*ω*= 2π

*/T*where

*T*is the OFC period). Equation (1) is in the form of an inverse discrete Fourier transform (IDFT). Therefore, the waveform

*a*(

_{R}*t*)’s shape is controlled by the

*R*coefficients and it is repetitive in nature with a period

_{n}*T*. Figure 7 shows how two different sets of

*R*produce two unique waveforms. For generality, the frequency axis is scaled to the OFC repetition rate, 1/

_{n}*T*, and the time axis is scaled to the OFC period,

*T*.

*m*(

_{n}*t*), to each comb line with the intent to produce a waveform

*a*(

_{m}*t*) that is continuous. We can define a frequency comb with modulated lines asAs mentioned previously, there are many sets of

*m*(

_{n}*t*) that can produce

*a*(

_{m}*t*) with bandwidth

*B*. At a minimum, a set of

*m*(

_{n}*t*) must be able to broaden each comb line to create a continuous spectrum. For example, the set of modulations that are the most electrically bandwidth efficient occur when the modulation bandwidth is equal to the comb spacing. The most inefficient (demanding) set of electrical modulations occur when one comb line is modulated at the full bandwidth

*B*and all other comb lines are off. A reasonable compromise between the two extreme cases takes place when the spectral slices partially overlap at their edges. The spectral-slice OAWG algorithm discussed in Section 5.2 provides a rigorous method to find many sets of modulations. By optimizing the parameters of the algorithm, the required bandwidths of the modulations and the computation time or latency are minimized.

*a*(

*t*), it is sufficient to check if the line-by-line sum of the product of the spectral slices and spectral filtering equals the target waveform’s spectrum,

*A*(

*ω*). Mathematically, this iswhere the individual spectral modulations,

*M*(

_{n}*ω*), are the Fourier transform (FT) of the temporal modulations,

*m*(

_{n}*t*), and

*H*(

_{n}*ω*) is the complex transmission function for the

*n*-th comb line through the multiplexer. When this equation is satisfied, the modulations will produce the target waveform. Such modulations are termed compatible with the spectral multiplexer. Correcting distortions due to the multiplexer filtering requires pre-emphasis of the modulations which is explained in more detail in the following section.

## 5. The dynamic-OAWG algorithm

*m*(

_{n}*t*), that when applied to a demultiplexed frequency comb, will exactly synthesize a continuous arbitrary optical waveform,

*a*(

*t*), with bandwidth

*B*. As Fig. 2(b)–2(d) illustrates, the algorithm provides continuous waveform generation in four steps: (1) temporally slice

*a*(

*t*) into subwaveforms

*a*(

_{k}*t*), (2) spectrally slice the

*k*-th subwaveform’s spectrum into

*n*slices, each corresponding to the

*n*-th line of the OFC, (3) determine the exact modulations needed for each subwaveform’s spectral slices using the spectral-slice OAWG (SS-OAWG) algorithm, and (4) temporally sum the

*n*-th spectral slice modulations for the subwaveforms. This last step provides continuous modulations for each spectral slice and the temporal slicing also permits computational parallelization of the algorithm.

*M*(

_{n}*ω*), that are a solution to Eq. (3). Since it involves computing the DFT of

*a*(

*t*), the SS-OAWG algorithm is restricted to periodic waveforms and prohibits continuous generation (i.e.,

*a*(

*t*) must be specified for its entire duration). To overcome this restriction, the dynamic-OAWG algorithm finds

*m*(

_{n}*t*) to generate

*a*(

*t*) in a series of smaller subwaveforms

*a*(

_{k}*t*). When the time-slicing is performed optimally, these computed modulations are the same as those provided by the infinite length SS-OAWG modulations.

### 5.1. Temporal slicing and spectral slicing

*β*= 1/2 (i.e., 50% excess width). For simplicity, the temporal slice filters have a uniform shape [i.e.,

*w*(

_{k}*t*) =

*w*(

*t*)] with a center-to-center spacing of

*T*. Therefore, when the temporal slicing is applied to the continuous waveform, it generates

_{S}*k*subwaveforms,

*a*(

_{k}*t*) =

*a*(

*t*)

*w*(

*t*−

*kT*) each with a duration

_{S}*T*. The summation of a set of slice filters must equal unity so that the summation across all

_{L}*a*(

_{k}*t*) is

*a*(

*t*). Temporal slice filters can overlap and this provides significant flexibility with the subwaveforms shape to produce compatible modulations.

*S*(

_{n}*ω*), the

*n*-th slice filter is associated with the

*n*-th line of the OFC. The spectral slice filters are assumed identical in shape and each is centered on a comb line [see Fig. 8(c) and 8(d)]. It is the application of slice filters in both the time and frequency domains that makes many solutions to Eq. (2) possible.

### 5.2. The spectral-slice OAWG algorithm and examples

*a*(

_{k}*t*), of duration

*T*is repetitive (with period

_{L}*T*) so that the modulations can be calculated. First, the subwaveform’s spectrum,

_{L}*A*(

_{k}*ω*), is computed using a DFT. Then the

*n*-th spectral modulation of the

*k*-th subwaveform is

*B*(

_{n}*ω*) =

*S*(

_{n}*ω*)/

*H*(

_{n}*ω*) which includes pre-emphasis for the spectral multiplexer [i.e.,

*H*(

_{n}*ω*)

^{−1}]. Substituting Eq. (4) into Eq. (3) yieldswhich holds true when the summation across a set of only the spectral slice filters is unity. The IDFT of the spectral modulations,

*M*(

_{k,n}*ω*), are the temporal modulations,

*m*(

_{k,n}*t*), necessary to generate a periodic version of

*a*(

_{k}*t*).

*B*(

_{n}*ω*) is finite where

*S*(

_{n}*ω*) is non-zero. Additionally, the impulse response of the corrected slice filter,

*b*(

_{n}*t*), predicts the shape of the temporal modulations. For example, using the rectangular spectral slice filter in Fig. 8(c) produces modulations bandlimited to the comb spacing. Because of the rectangular spectral shape, these modulations will have a sinc-like ringing in the time domain. Using the overlapping slice filter of Fig. 8(d) reduces the ringing in the time domain and even reduces the amount of pre-emphasis needed at the expense of slightly more bandwidth. Optimizing

*B*(

_{n}*ω*) is critical, since excessive amounts of pre-emphasis will increase optical losses through the OAWG device and reduce the effective dynamic range. Additionally, it will set limits on the temporal-slice spacing for high-fidelity continuous waveform generation.

*T*=

_{L}*T*) and (b) a waveform with a relatively long duration. All simulations use the spectral slice filter in Fig. 8(d), the defocused multiplexer shown in Fig. 6(c) and for simplicity, all 16

*R*are unity with zero phase (i.e., the input OFC has a flat spectral intensity and phase). Figure 9(a) shows the static-OAWG case where the computed comb line modulations are static. Figure 9(a.1) shows that the waveform

_{n}*a*(

_{k}*t*) is a repetitive waveform with a period equal to the OFC period,

*T*. The spectrum of

*a*(

_{k}*t*), Fig. 9(a.2), is obtained using a DFT and since the waveform is repetitive,

*A*(

_{k}*ω*) is a set of comb lines. The computed spectrum is spectrally sliced by the corrected slice filter [black and grey traces in Fig. 9(a.2)] and Fig. 9(a.3) presents the spectral modulations,

*M*(

_{k,n}*ω*), for the three center slices. Here, and as a general rule for the SS-OAWG algorithm, we actually use the corrected slice filter to spectrally slice

*A*(

_{k}*ω*) and therefore the calculated spectral modulations account for the shapes of both

*S*(

_{n}*ω*) and

*H*(

_{n}*ω*) in a single step. In the static-OAWG case [Fig. 9(a)], the corrected slice filters act as only isolation filters since the slice spectrum consists of just a single comb line. Figure 9(a.4) shows the temporal modulations [i.e., the IDFT of

*M*(

_{k,n}*ω*)]. Since

*a*(

_{k}*t*) is periodic, the modulations are static in time. Of course, previous static-OAWG theory produces the same set of modulations and does not require multiplication of

*B*(

*ω*) or application of the IDFT.

*T*> 1000

_{L}*T*—essentially infinite length). Figure 9(b.1) shows the waveform when its intensity is non-zero and Fig. 9(b.2) presents its nearly continuous spectrum (i.e., an infinite length waveform implies a continuous spectrum). Applying the pre-emphasis filters to spectrally slice

*A*(

_{k}*ω*) yields the spectral modulations,

*M*(

_{k,n}*ω*). Figure 9(b.3) shows the spectral modulations for the three center slices which are bandlimited by the pre-emphasis filter. The temporal modulations of the central comb line, Fig. 9(b.4), are time-varying and when summed with the other modulations, accurately reconstruct the waveform.

### 5.3. The complete dynamic-OAWG algorithm and examples

*a*(

*t*) into subwaveforms with duration

*T*and a center-to-center spacing

_{L}*T*using the temporal slice filter,

_{S}*w*(

*t*), so that each subwaveform is given by

*a*(

_{k}*t*) =

*a*(

*t*)

*w*(

*t*−

*kT*). Then it computes the comb line modulations,

_{S}*m*(

_{k,n}*t*), for each

*a*(

_{k}*t*) using the SS-OAWG algorithm. Finally, for each comb line, it temporally stitches together the subwaveform modulations. Thus, the continuous modulations for each spectral slice are realized bywhere rect is the rectangular function (i.e., normalized boxcar function where if |

*t*| ≤ ½, then rect(

*t*) = 1, otherwise rect(

*t*) = 0). The rectangular function is included to emphasize that the modulations

*m*(

_{k,n}*t*) produced by the SS-OAWG algorithm are repetitive with a period

*T*and they are mathematically time gated when used in the dynamic-OAWG algorithm. The calculated

_{L}*m*(

_{n}*t*) do not necessarily satisfy Eq. (3) since multiplication by the rect function in Eq. (6) occurs after

*B*(

_{n}*ω*) is applied to

*M*(

_{k,n}*ω*) and any new frequencies in

*M*(

_{n}*ω*) (e.g., from instantaneous jumps) are not included in the pre-emphasis (details in Section 5.4).

*T*and

_{S}*T*equal to the OFC period, is used on the long target waveform. (b) presents a more practical case where the overlapping temporal slice filter from Fig. 8(b), with

_{L}*T*= 2

_{S}*T*and

*T*= 2

_{L}*T*, is used to temporally slice

_{S}*a*(

*t*). In both cases, the multiplexer used to combine the spectral slices has the response shown in Fig. 8(d). The simulations show that the case (a) (i.e., rapid-update static-OAWG) requires extremely high-bandwidth modulation capability and spectrally flat multiplexers (e.g., star couplers) to avoid serious distortions. Alternatively, case (b) produces higher-fidelity waveforms while providing an optimization path for the dynamic-OAWG algorithm’s parameters.

*w*(

*t*) is applied to

*a*(

*t*). In case (a), the rectangular filter produces subwaveforms that are non-zero at their edges. Whereas case (b), the raised-cosine temporal slice filter, produces subwaveforms that overlap and also are zero at their edges. Figure 10(a.3) and 10(b.3) show some of the subwaveform modulations (real and imaginary values) calculated using the SS-OAWG algorithm after the rectangular function of Eq. (6) is applied. As mentioned previously, the DFT and IDFT in SS-OAWG algorithm require that both the subwaveforms and the modulations are periodic; the grey areas in Fig. 10(a.3) and 10(b.3) indicate one subwaveform period,

*T*. In case (a), the subwaveform modulations are constant across each subwaveform [similar to Fig. 9(a.4)]. The modulations are non-zero at the subwaveform edges, and therefore the rectangular gating function sets the modulations to zero.

_{L}*T*= 4

_{L}*T*) and overlapping

*w*(

*t*) drive the subwaveforms to zero at their edges. This produces much smoother

*m*(

_{k,n}*t*) that also approach zero at their edges [similar to Fig. 9(b.4)]. Figure 10(a.4) and 10(b.4) show

*m*

_{0}(

*t*), the real and imaginary center line modulations calculated using Eq. (6) (solid) and the ideal spectral slice modulations (dashed) obtained by applying the SS-OAWG algorithm on

*a*(

*t*) over a much longer window. For both cases,

*m*

_{0}(

*t*) has instantaneous jumps at the subwaveforms’ edges (every

*T*). However, for case (a), the jumps are much larger in magnitude and frequency while for case (b),

_{S}*m*

_{0}(

*t*) is nearly seamless. Figure 10(a.5) and 10(b.5) shows the Fourier transform of the three center-most comb line modulations. The modulations are not bandlimited although the modulations for case (b) occupy much less bandwidth than those of case (a). Figure 10(a.6) and 10(b.6) show the simulated dynamic-OAWG output waveform,

*d*(

*t*), for the two cases. The output waveform in case (a) has greater than 10% normalized energy error (NEE) while case (b) has only 1% NEE where the normalized energy error is defined aswhere

*d*(

*t*) is the generated waveform,

*a*(

*t*) is the target waveform, and

*μ*is a tuning parameter to minimize the NEE.

*m*(

_{k,n}*t*) are non-zero at the edges of the subwaveforms, then the gating in Eq. (6) introduces new frequencies that are not accounted for by the algorithm and distortions occur when the multiplexer filters these frequencies. That is, the modulations provided by the dynamic-OAWG algorithm only satisfy Eq. (3) for an all-pass multiplexer (i.e., no need for pre-emphasis). The distortions are difficult to quantify because they are both waveform and multiplexer dependent, and clustered near the edges of the subwaveforms. However, simulations show that the NEE is typically halved when

*T*doubles. In an optimized design of

_{S}*w*(

*t*) and

*S*(

_{n}*ω*), the dynamic-OAWG algorithm converges to the modulations determined using the spectral-slice OAWG of the infinite-length

*a*(

*t*).

### 5.4. Optimization of the dynamic-OAWG algorithm

*w*(

*t*), the waveform fidelity is compared for four different corrected slice filters [see Fig. 11(c)] to understand how they impact fidelity and latency. In general, waveform errors occur at the edges of each temporal subwaveform because this is where the instantaneous jumps in the modulations occur and are filtered. The fidelity is improved by increasing

*T*or

_{S}*T*, optimizing

_{L}*S*(

_{n}*ω*) to generate a flatter corrected slice filter with smooth roll-off,

*B*(

_{n}*ω*), optimizing the multiplexer for greater spectral overlap, and forcing the time slice filters to slowly roll-off to zero at their edges.

*w*(

*t*) cases in Fig. 11(a), the NEE drops by half (−3 dB) when the latency doubles. This is because the combined modulations have significant energy at the subwaveform edges. In turn, the subwaveform modulations are large and produce discontinuities when combined together. These discontinuities are filtered and create distortions located near the subwaveform window edges. The corrected slice filter does not accommodate these new frequencies and the improvement in fidelity versus latency is mainly the result of fewer distortions or glitches across the waveform.

*w*(

*t*) that forces the subwaveforms to zero at their edges and therefore the corresponding subwaveform modulations approach zero at their edges. Here, optimizing the impulse response of the corrected slice filter plays a strong role in reducing the waveform errors. Generally, corrected slice filters that have wider bandwidths and roll off smoothly have impulse responses that are more localized in time.

*β*= 1/2 temporal slice filter and corrected slice filter C. The right waveform uses the rect temporal slice filter and corrected slice filter C. Waveforms are computed at 60

*T*ps latency. The left waveform has 1% NEE and there is a noticeable difference between the generated waveform

*d*(

*t*) and the target waveform

*a*(

*t*). The most significant errors are localized near the subwaveform transition points. The right waveform has only 0.0001% NEE and it is indistinguishable from the target on the scale shown.

## 6. Dynamic-OAWG and an analogy with OAWM

*I/Q*modulation), determined with DSP pre-processing, are applied to each OFC comb line in a parallel manner. The spectrally broadened comb lines are coherently combined using a multiplexer with spectrally overlapping adjacent passbands, producing the arbitrary optical waveform. The spectrum of the resulting waveform is continuous with the phase and amplitude fully specified across the entire bandwidth. If the modulation is continuously applied, the waveform's temporal length can be infinite.

*I/Q*signals using DSP. If the data is recorded continuously, the spectral resolution is limited only by the reference OFC stability and digital coherent receiver performance.

8. 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]

## 7. Conclusion

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), Paper OThS1.

## Acknowledgments

## References and links

1. | K. Takiguchi, K. Okamoto, T. Kominato, H. Takahashi, and T. Shibata, “Flexible pulse waveform generation using silica-waveguide-based spectrum synthesis circuit,” Electron. Lett. |

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

3. | P. J. Delfyett, S. Gee, C. Myoung-Taek, 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. |

4. | N. K. Fontaine, R. P. Scott, J. Cao, A. Karalar, W. Jiang, K. Okamoto, J. P. Heritage, B. H. Kolner, and S. J. Yoo, “32 Phase X 32 amplitude optical arbitrary waveform generation,” Opt. Lett. |

5. | R. P. Scott, N. K. Fontaine, J. Cao, K. Okamoto, B. H. Kolner, J. P. Heritage, and S. J. B. Yoo, “High-fidelity line-by-line optical waveform generation and complete characterization using FROG,” Opt. Express |

6. | 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 |

7. | 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. |

8. | 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 |

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

10. | N. K. Fontaine, D. J. Geisler, R. P. Scott, T. He, J. P. Heritage, and S. J. B. Yoo, “Demonstration of high-fidelity dynamic optical arbitrary waveform generation,” Opt. Express (submitted for publication). [PubMed] |

11. | N. K. Fontaine, R. P. Scott, C. Yang, D. J. Geisler, J. P. Heritage, K. Okamoto, and S. J. B. Yoo, “Compact 10 GHz loopback arrayed-waveguide grating for high-fidelity optical arbitrary waveform generation,” Opt. Lett. |

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 |

13. | T. He, N. K. Fontaine, R. P. Scott, D. J. Geisler, J. P. Heritage, and S. J. B. Yoo, “Optical arbitrary waveform generation based packet generation and all-optical separation for optical-label switching,” IEEE Photon. Technol. Lett. |

14. | D. J. Geisler, N. K. Fontaine, T. He, R. P. Scott, L. Paraschis, J. P. Heritage, and S. J. Yoo, “Modulation-format agile, reconfigurable Tb/s transmitter based on optical arbitrary waveform generation,” Opt. Express |

15. | 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. |

16. | J. T. Willits, A. M. Weiner, and S. T. Cundiff, “Theory of rapid-update line-by-line pulse shaping,” Opt. Express |

17. | R. G. Lyons, |

18. | R. Trebino, |

19. | M. K. Smit and C. Van Dam, “PHASAR-based WDM-devices: Principles, design and applications,” IEEE J. Sel. Top. Quantum Electron. |

20. | K. Okamoto, “Recent progress of integrated optics planar lightwave circuits,” Opt. Quantum Electron. |

21. | N. K. Fontaine, Y. Jie, J. Wei, D. J. Geisler, K. Okamoto, H. Ray, and S. Yoo, “Active arrayed-waveguide grating with amplitude and phase control for arbitrary filter generation and high-order dispersion compensation,” in |

**OCIS Codes**

(320.0320) Ultrafast optics : Ultrafast optics

(320.5540) Ultrafast optics : Pulse shaping

(320.7100) Ultrafast optics : Ultrafast measurements

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: July 12, 2010

Revised Manuscript: August 6, 2010

Manuscript Accepted: August 7, 2010

Published: August 17, 2010

**Citation**

Ryan P. Scott, Nicolas K. Fontaine, Jonathan P. Heritage, and S. J. B. Yoo, "Dynamic optical arbitrary waveform generation and measurement," Opt. Express **18**, 18655-18670 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-18-18655

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### References

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- Z. Jiang, D. E. Leaird, and A. M. Weiner, “Line-by-line pulse shaping control for optical arbitrary waveform generation,” Opt. Express 13(25), 10431–10439 (2005). [CrossRef] [PubMed]
- P. J. Delfyett, S. Gee, C. Myoung-Taek, 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]
- N. K. Fontaine, R. P. Scott, J. Cao, A. Karalar, W. Jiang, K. Okamoto, J. P. Heritage, B. H. Kolner, and S. J. Yoo, “32 Phase X 32 amplitude optical arbitrary waveform generation,” Opt. Lett. 32(7), 865–867 (2007). [CrossRef] [PubMed]
- R. P. Scott, N. K. Fontaine, J. Cao, K. Okamoto, B. H. Kolner, J. P. Heritage, and S. J. B. Yoo, “High-fidelity line-by-line optical waveform generation and complete characterization using FROG,” Opt. Express 15(16), 9977–9988 (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(8), 463–467 (2007). [CrossRef]
- 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]
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- N. K. Fontaine, D. J. Geisler, R. P. Scott, T. He, J. P. Heritage, and S. J. B. Yoo, “Demonstration of high-fidelity dynamic optical arbitrary waveform generation,” Opt. Express (submitted for publication). [PubMed]
- N. K. Fontaine, R. P. Scott, C. Yang, D. J. Geisler, J. P. Heritage, K. Okamoto, and S. J. B. Yoo, “Compact 10 GHz loopback arrayed-waveguide grating for high-fidelity optical arbitrary waveform generation,” Opt. Lett. 33(15), 1714–1716 (2008). [CrossRef] [PubMed]
- 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), Paper OThS1.
- T. He, N. K. Fontaine, R. P. Scott, D. J. Geisler, J. P. Heritage, and S. J. B. Yoo, “Optical arbitrary waveform generation based packet generation and all-optical separation for optical-label switching,” IEEE Photon. Technol. Lett. 22(10), 715–717 (2010). [CrossRef]
- D. J. Geisler, N. K. Fontaine, T. He, R. P. Scott, L. Paraschis, J. P. Heritage, and S. J. Yoo, “Modulation-format agile, reconfigurable Tb/s transmitter based on optical arbitrary waveform generation,” Opt. Express 17(18), 15911–15925 (2009). [CrossRef] [PubMed]
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- N. K. Fontaine, Y. Jie, J. Wei, D. J. Geisler, K. Okamoto, H. Ray, and S. Yoo, “Active arrayed-waveguide grating with amplitude and phase control for arbitrary filter generation and high-order dispersion compensation,” in 34th European Conference on Optical Communication (ECOC 2008), Technical Digest (CD) (IEEE, 2008), 1–2. http://dx.doi.org/10.1109/ECOC.2008.4729152

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