## Abstract

We demonstrate a new photonically assisted reconfigurable radio-frequency waveform generator. The setup is based on phase modulating a multi-wavelength pulse source and subsequent compression in a dispersive medium. Under the appropriate conditions, we show that the photodetected electrical signal is broadband and coherent. Specifically, we show that this system allows for the synthesis of a reconfigurable finite-impulse-response filter where the number of filter taps is given by the number of wavelengths available from the multi-wavelength source and the reconfiguration is determined simply by their power and wavelength separation. We also show that this technique allows for time-multiplexing the synthesized waveforms, thus leading to an effective switching speed fixed by the clock rate. In particular, we show transitions between synthesized waveforms with a frequency content > 60 GHz in periods shorter than 100 ps.

©2009 Optical Society of America

## 1. Introduction

In the last years there has been a growing interest in the development of photonically assisted radio-frequency arbitrary waveform generators (RF-AWGs) (see .e.g., [1–13]). The inherent broadband nature of photonics promises enhanced capabilities for synthesizing electronic signals over spectral regions that are very difficult to reach by conventional electronics, such as the ultra-wideband domain (3.1-10.6 GHz) [4,8], the mm-wave regime (~30-100 GHz) [2, l 3] or even the terahertz range (~0.1-3 THz) [1, 13]. These synthesizers would be a powerful tool for a broad range of applications, including the testing of high-speed optoelectronic equipment, radar, sensing, and communications [14–17]. Other important aspects of working directly in the optical domain are the possibility of transporting the signal through a fiber link to a remote location [14], or the flexibility in the reconfiguration of the synthesized waveforms [15, 16].

Many interesting optical solutions exist that provide the desired wideband capabilities. Among these, AWGs based on pulse shapers implemented either in bulk [1–4,11–13] or arrayed waveguide grating technology [7] lead a better performance in terms of reconfigurability and complexity of the achievable waveform. In this case, the waveform is tailored through Fourier synthesis of a coherent [1–4, 13] or even incoherent optical source [11, 12]. Usually, since the generated pulses have temporal features on the order of ps, the maximum achievable bandwidth is ultimately limited by the photodetector response time. A significant drawback of these devices is that they do not offer a reconfiguration speed in the GHz regime, mainly due to the limited response time of the spectral shaper element (usually based on liquid crystal or thermo-optic modulation).

Since it is unlikely that this switching technology will approach the GHz regime in the near future, researchers have focused their efforts on implementing different time-multiplexing configurations [18–22]. We mention, for example, the alternate switching of an optical frequency comb generator (OFCG) that is shaped by two different Fourier processors [21], or to switch two different combs centered at different wavelengths using a single spatial light modulator [20]. Although this last solution generates alternating waveforms centered at different wavelengths, it is not a problem for RF-AWG, where the optical signal is detected in intensity. It is important to note that in the previous setups, the switching capabilities are not achieved by the line-by-line shaping processing, but owing to the multiplexing scheme.

In this work, we propose and demonstrate an alternative, reconfigurable and rapidly switchable RF-AWG. It is based on the generation and compression of a multi-wavelength pulse train [23]. The generation of ultrashort light pulses with an electro-optic phase modulator (EOPM) and subsequent intensity conversion in a dispersive medium is a well established technique [24–27]. In a recent work, this concept was extended to multiple wavelengths [23], proving to be a very useful and reliable approach for generating a time-interleaved pulse train with alternate carriers. The key is to note that the time separation between pulses at two wavelengths can be precisely controlled by the combined effect of dispersion and wavelength separation. Here, we study the suitability of this technique for RF-AWG. Specifically, we show that when the photodiode bandwidth is smaller than any possible beating from the spectra of adjacent phase-modulated lasers, the generated electrical signal is given by a sum of weighted and delayed replicas of the compressed intensity pulses. The tap weights and delays are controllable by the power and wavelength spacing of the lasers. In mathematical terms, this means that we can implement a finite impulse response (FIR) filter over the shape of the compressed pulse intensity. Essentially, this is an incoherent scheme in the sense that intensities are summed at the photodetector, but which provides a coherent electrical signal. Since the shaping relies on the tunability of the laser sources, no active shaping element is required, leading to a simple yet powerful technique to perform an RF-AWG. We note that the complexity of the filter synthesis is limited by the number of taps and the fact that they can be only positive. Nevertheless, we show that several interesting waveforms can be created with only four taps. Finally, we show that this scheme also offers the possibility for time-multiplexing of waveforms that are synthesized using different arrays of lasers.

The remainder of this paper is structured as follows. In Section 2 we provide the basic theory and the experimental results for our static RF-AWG. We investigate the effects of the finite linewidth and analyze the limits of the technique. In Section 3, we describe the experimental verification for the time-multiplexing operation. Finally, we summarize the work and present the main conclusions in Section 4.

## 2. Multi-wavelength pulse compression and static waveform generation

#### 2.1 Heuristic description of the physical principle

Consider a multi-wavelength pulse source like the one shown in the cartoon of Fig. 1(a) . Let us assume that each of the color pulses travels, possibly although not necessarily, with distortion through a device that delays each pair of two neighboring pulses in a quantity proportional to their carrier separation. Later on, the intensities are summed at the photodetector. It is clear that by controlling the wavelength separation (and therefore the delay) and the power of each light pulse, one can synthesize the received electrical signal. The effect of the photodiode’s finite bandwidth is to broaden the received intensity pulse.

This is exactly the scheme that we proposed in [6] and verified in [11, 12]. There, the array of wavelengths was a continuous distribution (a broadband amplified spontaneous emission source), which was modulated by an external modulator and the delay element was simply a first-order dispersive medium. The continuous nature of the source allowed us to identify the reshaping process as a “frequency-to-time mapping”. The intensity summation was due to the uncorrelation of the different spectral components. However, this comes at the expense that the detected signal has a low signal-to-noise ratio (SNR), therefore limiting the range of applications of this RF-AWG. Intuitively, the origin of this noise comes from the random beating of the different spectral components at the photodetector. In consequence, by reducing the photodetection bandwidth, the SNR increases. Further details and calculations can be found in [28].

Let us now consider the cartoon of Fig. 1(a) but implemented by the setup of Fig. 1(b). Here, an array of tunable lasers (not necessarily equally spaced in wavelength) constitutes the light source, which is phase modulated by a sinusoid signal and carved by an external amplitude modulator. The reshaping device is another dispersive medium properly selected to compensate for the chirp imparted by the phase modulator, thus producing a compressed pulse for each of the carriers. Again, the delay between two consecutive pulses is given by a quantity proportional to their carrier wavelength separation and the dispersion amount [23]. The intensity summation is achieved by separating the wavelengths in such a way that any possible beating between the generated sidebands from two different carriers disappears. We have used the same concept to induce an incoherent superposition (sum in intensity) for the temporal Talbot effect [29]. The signal detected by the photodiode is not strictly coherent owing to the finite laser linewidth. However, as shown in the next subsection, this effect is expected to be minimal and can be practically disregarded when dealing with narrow-linewidth lasers. It is worth mentioning that the same trick of wavelength separation can be used in [11, 12] to increase the SNR [30], although at the expense of increasing the detected pulse width [28]. However, please note that in the proposed technique, thanks to the phase-to-intensity conversion, the pulse impinging at the photodiode is actually shorter than the pulse carver gate, leading to a broadband and coherent electrical signal. Finally, the synthesis of the generated electrical pulse is performed by controlling the wavelength separations and powers of the tunable lasers.

#### 2.2 Theory

Here we provide a mathematical model that describes the above heuristic picture. We follow the approach developed in [23], but we include now the effect of the lasers’ linewidth. We consider an array of *N* tunable lasers phase-modulated by a time lens and carved by an intensity modulator. The complex electric field before the dispersion stage is given by

*K*is the chirping rate of the time lens. For a time lens implemented in an electro-optic phase modulator $K=-4\Delta \theta {\pi}^{2}{f}_{r}^{2}$, with $\Delta \theta $ denoting the modulation index and ${f}_{r}$ the repetition rate. For simplicity, we will assume an aberration-free time lens. ${\omega}_{n}$ is the carrier frequency of the

*n*th laser, ${P}_{n}$ its power, and ${\varphi}_{n}(t)$ is a real random phase function accounting for the possible phase noise. The linewidth shape for the

*n*th laser is given by the inverse Fourier transform (with respect to the variable τ) of the autocorrelation function $\u3008\mathrm{exp}[j\{{\varphi}_{n}^{}(t)-{\varphi}_{n}^{}(\tau -t)\}\u3009$, with $\u3008\u3009$ denoting time average. Assuming that the propagation through the linear dispersive medium is given by the transfer function $H(\omega )=\mathrm{exp}[j\{{\Phi}_{1}(\omega -{\omega}_{0})+{\Phi}_{2}{(\omega -{\omega}_{0})}^{2}/2\}]$, with ${\Phi}_{1}$ and ${\Phi}_{2}$ being respectively the group-delay and group-delay-dispersion coefficients, the output field becomes

*t’*is expressed as $t\text{'}=t-{\Phi}_{1}$; ${\alpha}_{n}=\sqrt{{P}_{n}}\mathrm{exp}[j({\Phi}_{2}{\omega}_{n}^{2}/2-{\omega}_{n}{\omega}_{0}{\Phi}_{2})]$, and ${a}_{n}(t)={\displaystyle \int {M}_{n}(\omega )}\mathrm{exp}[j{\Phi}_{2}{\omega}^{2}/2]\mathrm{exp}[-j\omega t]d\omega $, with ${M}_{n}(\omega )$ being the Fourier transform of $m(t)\mathrm{exp}[-j{\varphi}_{n}(t)]$. The delay is given byTherefore, the laser arbitrarily marked as “0” sets the time origin. The delays for the rest of the lasers are measured with respect to this time reference. Giving one step further we can calculate the instantaneous optical intensity

*B*is the photodiode bandwidth and ${\sigma}_{\alpha ,f}$ is the spectral width of ${a}_{n+1}^{*}(t-{t}_{n+1}){a}_{n}^{}(t-{t}_{n})$. Therefore, ${\sigma}_{\alpha ,f}$ is roughly given by the optical spectrum width of $m(t)$. Under these conditions, we get that the photodetected intensity becomes ${i}_{out}(t\text{'})\approx ({\displaystyle {\sum}_{n=0}}{P}_{n}{\left|{a}_{n}(t\text{'}-{t}_{n})\right|}^{2})\otimes {h}_{PD}(t\text{'})$, with ${h}_{PD}(t)$ being the impulse response of the photodiode. But let us provide a further inspection into the structure of ${a}_{n}^{}(t)$, and in particular an assessment of the effects of the laser linewidth. We have

Equation (8) is a key result of the present work. It states that under the incoherent regime established by Eq. (5), there is a linear relation between the photocurrent and the compressed intensity pulse shape of a single wavelength. The electrical impulse ${i}_{out}(t\text{'})$ can be tailored through the power of the tunable lasers, ${P}_{n}$, and their wavelength spacing [leading to a time delay ${t}_{n}$ provided by Eq. (3)]. This defines an FIR filter with positive coefficients only, whose complexity is directly related to the number of taps *N*. Using *N* tunable laser sources, the filter can be easily reconfigured without altering the external modulation or dispersive medium. Although this is an incoherent operation, in the case where the laser linewidth is smaller than the optical spectrum of the carved pulse the generated electrical signal ${i}_{out}(t)$ is highly coherent because only deterministic functions contribute to its shape.

It is important to realize that despite the semblance with a microwave photonic filter (MWPF) with *N* taps [33,34], the device we are proposing is not an incoherent MWPF. The subtle yet important difference in our configuration is the introduction of the time lens. In the deep modulation regime, this device makes it impossible to establish a linear relation between the output and input electrical signals (a sinusoid in this case). In our configuration however, the linear relation is established between the output electrical signal and the compressed optical intensity at a single wavelength. However, we recognize that since our system defines an FIR with positive taps, the same techniques developed in the microwave photonics literature [33] for achieving a particular filter response can be easily adapted for our configuration [35,36]. Furthermore, although here an FIR operation with positive taps only has been performed, it is not difficult to envision an upgrade of our system to include negative taps by, for example, using balanced photodetection or cross-gain modulation. Similar solutions have been previously reported to implement incoherent MWPFs with negative taps [16, 17, 33].

#### 2.3 Complexity and resolution limits of the technique

The minimum temporal feature that can be achieved in our RF waveform generator depends on the temporal duration of the compressed pulse. For deep modulation indices (>> π rad) and high clock frequencies (~ 10 GHz), it is possible to obtain few ps and even sub-ps short pulses [23–27], providing few tens to hundreds of GHz RF bandwidth to be shaped.

On the other hand, it is clear that the complexity of the achievable electrical waveform is directly related with the complexity of the FIR filter we can implement, which in turn is given by the number of optical taps. A fast inspection of Eq. (3), taking into account the condition provided by Eq. (5), shows that the maximum number of taps that one can generate is very low if one desires to restrict the pulses to fall inside the period range of the modulator, *T*. However, if we relax this condition to [23]

*P*denotes a large integer number and $0\le \delta \le 1$, then we can still overlap the pulses inside a period, owing to the periodicity of the pulse carve train and phase modulation. In this case, the limiting factor becomes the available bandwidth of the dispersive medium. The reason is that the deeper the phase modulation, the broader the optical spectrum, and thus the wavelengths must be placed further apart in order to satisfy the incoherent operating regime. Consequently, there is a tradeoff in our configuration between the maximum achievable bandwidth of the electrical signal and the filtering complexity. If we call $\Delta \Omega $ the available optical bandwidth in the dispersive element, the maximum number of taps that can be implemented in the filter is

*B*~50 GHz, ${f}_{r}\approx 10\text{\hspace{0.17em}}$GHz and a modulation index of

*π*rad, the maximum number of taps we can implement is ~16. Therefore, compared with coherent optical frequency comb techniques, where the complexity of the generated waveform is directly related to the number of comb lines, our synthesizer has a reduced capability in terms of RF waveform complexity because typical frequency comb generators can provide the shaping of tens [7,37] to more than one hundred [38] lines (if nonlinear effects are involved). However, we next show that even using a limited number of positive taps, very interesting waveforms can be achieved.

#### 2.4 Experimental results

Figure 2 shows the scheme implemented for the generation of static (although reconfigurable) RF signals. 2 tunable lasers in the C-band and 2 DFB lasers with a limited tunable range are used as light sources. The pulse carving operation is implemented by an electro-optic modulator (EOM) driven by a periodic signal at 12 GHz. It is biased to produce a nearly 45% duty-cycle pulse train. The pulse train is optically phase modulated by an EOPM driven by an amplified sinusoidal electrical signal. From an analysis of the dependence of the relative optical power in the spectrum sidebands with the voltage value applied on the EOPM, we measured the ${V}_{\pi}$ parameter at 12 GHz. From this parameter, we inferred a maximum achievable value of $\Delta \theta \approx $0.7π rad. We used a second EOM driven by a 12.5 Gb/s pulse pattern generator (PPG) to reduce the repetition rate of the pulse train without altering the clock frequency (therefore keeping the chirping rate intact). We used here the code “1010” (leading to a 6 GHz repetition rate). The expected and measured characteristics of the compressed pulse at a single wavelength after propagation in a dispersive single-mode fiber (SMF) of 3.2 km length are illustrated in Fig. 3 . We observed a compressed pulse with 10.9 ps pulse width [Fig. 3(d)], in close agreement with the expected value of 11.9 ps [Fig. 3(a)]. All the time measurements are taken with a 55 GHz sampling scope with no averaging and 400 ms persistence time. The slight discrepancy could be due to an underestimation of the modulation index, which can be confirmed from the fact that the measured optical spectrum [Fig. 3(f)] is slightly broader than the expected [Fig. 3(c)]. According to Eq. (8), the pulse profile illustrated in Fig. 3 (d) represents our unit cell ${I}_{0}(t)\otimes {h}_{PD}(t)$. Note that since our technique is based on the intensity profile of the achieved pulse, it is irrelevant whether the optical pulse is chirp free or not. The optical spectrum is measured after the modulation stage using an optical spectrum analyzer (OSA) with 0.8 pm resolution (Apex AP2443B), which allows for observing the comb structure with a line spacing of 6 GHz. We did not appreciate significant differences neither in the intensity pulse shapes nor optical spectra for different carriers in the 1525-1560 nm range. The small pedestal on the compressed pulse is due to the time lens aberrations [23, 25], but it does not represent a significant issue for the particular electrical waveforms synthesized in this work. If needed, it could be removed by introducing an extra EOPM to compensate for the non-quadratic phase profile in the chirping [23].The simulated RF spectrum of the signal is displayed in Fig. 3(b). A bell-shape signal appears with a frequency content higher than 60 GHz at the −20 dB level.

We emphasize that unlike with frequency-comb based shaping, where the amplitude and/or phase of the individual lines is manipulated, our proposal is an incoherent scheme and only relies on the spectral position and the global power of the comb. By switching on the remaining lasers, the same pulse profile appears shifted in time by a quantity proportional to the wavelength separation and dispersion.

We now proceed to show some examples illustrating the capabilities of the technique. Basing on Eq. (3), and taking into account the 3.2 km fiber dispersion (${\Phi}_{2}=-69\text{\hspace{0.05em}}p{s}^{2}$), we are able to achieve a relative delay per wavelength separation of 0.43 ps/GHz. Owing to the incoherent regime dictated by Eq. (5), the minimum achievable delay without producing a beating should be ~57 ps. Although this is a large delay compared to the period, we remind that thanks to the periodicity of the compressed train, by separating further apart each comb, several unit cell pulses may overlap inside a period and electrical waveforms can be composed with an accuracy as good as 0.43 ps/GHz. Additionally, we measured the optical bandwidth of the SMF with a time-of-flight technique (the available bandwidth is defined as the spectral region where the third-order-dispersion effects can be disregarded), leading to ~2.5 THz. Basing on Eq. (12) and taking into account the previously mentioned parameters, with our setup we can introduce a maximum number of 18 taps. The small number of taps here used is limited by the tunable lasers available in our facilities.

We start with the formation of a square pulse. Figure 4 (a)-(d)
show the intensity profiles achieved when only the corresponding optical tap is switched on. Their respective optical spectra are shown in (e)-(h) in false color. The incoherent regime is guaranteed because we avoided spectral overlapping between the frequency combs, as illustrated by the optical spectrum measured in Fig. 4 (i). Note that these spectra show that the overlapping inside the waveform period comes from individual unit cell pulses whose *P* value from Eq. (11) is not the same (the position of each comb is not increasing linearly with the wavelength). This is not a problem because the meaningful quantity is the relative delay inside the period, $\delta T$, and not the absolute value ${t}_{n}$. We selected the wavelength spacing and power values so that when the four lasers are on, these individual intensity pulses generate a square-wave pulse. We have measured a pulse width of 40.4 ps duration, and a rise and fall time of 7.3 and 6.1 ps, respectively. We note that the summation is not exactly linear. This is due to gain redistribution in the optical amplifiers when all the lasers are on. It is thus a deterministic effect that can be taken into account for the waveform synthesis problem. For completeness, Figs. 4(k) and (k,l) show the raw data superimposed in time and a snapshot of the electrical pulse sequence. Typical peak-to-peak amplitudes without electrical amplification are ~100 mV when ~5 dBm optical power is launched to the photodiode.

Figure 5 shows two more examples involving the generation of a triplet and a burst pulse with increasing intensity. Again, we see that due to the gain redistribution, the intensity summation for all the taps is not exact.

## 3. Time-multiplexing of radio-frequency waveforms

The generated waveforms in the previous section are easily reconfigurable by changing the wavelength separation and power of the individual lasers. In this Section we aim to provide a functionality that permits the rapid change from one waveform to another in a very short transition time (<100 ps). To this aim, we have focused ourselves in a time-multiplexing scheme. This technique has been used previously for optical [18, 21] and radio-frequency [19,20] waveforms generated from spectral shaping of optical broadband signals. In particular, we have implemented the setup shown in Fig. 6 . It is basically the same setup as Fig. 2, but we have added an extra array of lasers and modulator (EOM2). While the previous EOM was driven by the data output from the PPG, the new one is driven with the data bar output. This, together with the extra tunable optical delay, allows us to ensure that when one EOM is on the other is off and vice versa. Therefore, the array of lasers that are modulated by the pulse carver is different every period, leading to transitions between the synthesized waveforms at the clock rate (12 GHz).

From a practical perspective, one could use the same array of lasers to drive both modulators and then use individual amplitude controllers. However, we have noted that due to the finite extinction ratio of the EOMs driven by the PPG, a remaining optical signal appears even when the EOM is in the off state. This effect leads to beating noise when the laser wavelengths are close each other. Therefore, each waveform was synthesized using different laser arrays.

Figure 7 shows an example of the time multiplexing capabilities of our scheme. We have implemented two different filters with three taps each. Figures 7 (a) and (b) show the intensity pulse gates from each modulator when a code “1010” is applied on the PPG. Figures 7 (c) and (d) represent the achieved waveforms from the taps coming from modulators 1 and 2, respectively. We see that we are able to synthesize the multi-wavelength pulse train in order to keep the taps inside a period. Figures 7 (e) and (f) show a zoomed version of the waveforms shown in (c) and (d), respectively. The waveforms are multiplexed when both arrays of lasers are on, as illustrated in Figs. 7 (g) and (h). This is a pulse burst with increasing and decreasing intensity.

Finally, Fig. 8 shows another example using a different code length (“1000”). A single tap coming from EOM1 produces a compressed pulse, while three taps switched in EOM2 lead to a continuous high-repetition pulse train. We believe this type of rapidly changing pulses could be useful to evaluate the response of electronic equipment to rapid transients of broadband RF signals.

## 4. Summary and conclusions

We have studied the capabilities of multi-wavelength time-lens compression for the development of an incoherent RF-AWG. The incoherent operation is achieved by avoiding spectral overlapping from different frequency combs. In this way, the interference is lost and the electrical signal generated in the photodetector is the sum of the optical powers from the individual wavelengths, leading to a highly coherent waveform if the linewidth of the individual optical sources is narrow. In this incoherent regime, the RF signal can be precisely and straightforwardly tailored through an FIR filter easily reconfigurable through the lasers’ wavelength separation and power, therefore avoiding the use of any active spectral shaper element. The main limitation of our scheme is the requirement of *N* tunable lasers for implementing an *N-*tap FIR filter. However, we believe that our setup constitutes an interesting alternative to RF-AWG implemented by coherent frequency comb technology when the desired electrical target waveforms are not extremely complex (such as squares, triangles or high-speed bursts, for example).

Additionally, we have shown that this technique enables the time-multiplexing of different waveforms. This allows for fast transitions (at GHz rates) between alternating waveforms, a feature that is gaining increasing attention. We have reported switching at 12 GHz rate between waveforms containing a bandwidth higher than 60 GHz. We believe this technique could be used, for example, for testing the behavior of broad-bandwidth optoelectronic components to fast transitions.

Finally, we would like to mention that this technique might benefit from the recent advances in WDM photonic integrated circuits [39], in particular by the possibility of integrating in a single photonic chip the transmitter part from the scheme of Fig. 6 extended to several arrays of *N* lasers.

## Acknowledgments

This work was supported in part by the National Science and Engineering Research Council (NSERC) of Canada. Victor Torres-Company acknowledges funding from the Spanish Ministry of Science and Innovation, and the Spanish Foundation for Science and Technology (FECYT) through a postdoctoral fellowship.

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