Abstract
Sincshaped Nyquist pulses possess a rectangular spectrum, enabling data to be encoded in a minimum spectral bandwidth and satisfying by essence the Nyquist criterion of zero intersymbol interference (ISI). This property makes them very attractive for communication systems since data transmission rates can be maximized while the bandwidth usage is minimized. However, most of the pulseshaping methods reported so far have remained rather complex and none has led to ideal sinc pulses. Here a method to produce sincshaped Nyquist pulses of very high quality is proposed based on the direct synthesis of a rectangularshaped and phaselocked frequency comb. The method is highly flexible and can be easily integrated in communication systems, potentially offering a substantial increase in data transmission rates. Further, the high quality and wide tunability of the reported sincshaped pulses can also bring benefits to many other fields, such as microwave photonics, light storage and alloptical sampling.
Introduction
In currently deployed optical networks, wavelength division multiplexing (WDM) is used to enhance the carrier capacity of optical fibres. However, since the data rate in optical networks increases by close to 29% per year^{1}, new approaches are being developed^{2}. The bulk of these approaches consists in increasing the spectral efficiency of optical links. Using multilevel modulation formats and polarization multiplexing, the spectral efficiency can be increased from 0.8 to several bit s^{−1}Hz^{−1} (refs 3, 4, 5). However, such schemes drastically increase the requirements on electrical signal processing and are typically accompanied by higher energy consumption. To keep pace with the growing demand, a data rate of 1 Tbit s^{−1} per channel together with high spectral efficiency has been envisaged for the next decade^{6}. Even with parallelization, these data rates are beyond the limits of current digital signal processing, and the resulting baud rate exceeds the possibilities of current electronic circuits^{7}. A possible solution is the combination of several lowerrate channels with high spectral efficiency into a Tbit s^{−1} ‘superchannel’, which can be routed through the existing optical networks as a single entity^{8}. Such an aggregation can be achieved in the frequency or time domain^{9}. In orthogonal frequencydivision multiplexing (OFDM), a superchannel consisting of a set of subcarriers is generated. Each subcarrier exhibits a sincshaped spectrum and can therefore be spaced at the baud rate without interchannel interference. With OFDM, a data rate of 26 Tbit s^{−1} and a net spectral efficiency of 5 bit s^{−1} Hz^{−1} have been demonstrated^{10}. Similarly, for Nyquist transmission, the symbols are carried by Nyquist pulses^{11} that overlap in the time domain without ISI. Recently, a 32.5Tbit s^{−1} Nyquist WDM transmission with a net spectral efficiency of 6.4 bit s^{−1} Hz^{−1} has been shown^{12}. Compared with OFDM, Nyquist pulse shaping has several unique advantages as it reduces the receiver complexity^{13,14}, is less sensitive to fibre nonlinearities^{14}, requires much lower receiver bandwidths^{15} and leads to lower peaktoaverage power ratios^{16}.
A general expression in the time domain for the amplitude waveform of Nyquist pulses is^{17,18}:
where τ_{p} is the pulse duration between zero crossings and β is known as a rolloff factor^{17}, which is in the range 0≤β≤1. Among the class of Nyquist pulses^{11}, the sincshaped pulse is of particular interest owing to its rectangular spectrum^{17} and zero rolloff. This allows minimizing the guard band between optical channels. Theoretically, for a sincpulse Nyquist transmission, each symbol consists of a timeunlimited sincpulse. However, since causality makes it impossible, periodic pulses are typically used in every experimental demonstration of Nyquist pulse transmission^{12,13,14,15,16,17,18,19,20}. Such transmission systems rely on multiplexing and modulation techniques. A possible scheme is shown in Fig. 1. Nyquist channels can be multiplexed in time domain; this is designated as orthogonal timedivision multiplexing (TDM)^{18,21,22}. The generated sequence is split into N channels, which are then delayed and modulated to transport the channel corresponding data. This requires N modulators, with N being the number of branches or the number of timedomain channels. However, compared with a direct modulation, the baud rate of each modulator is N times reduced. This drastically relaxes the requirements on modulators and electronics. In addition, timedomain channels can be multiplexed at different wavelengths; this is designated as Nyquist WDM^{8} where pulses can be generated and modulated for each carrier. Since higherorder modulation formats, multiplexing, transmission and demultiplexing of Nyquist pulses have already been shown elsewhere^{12,13,14,15,16,17,18}, here the focus is placed on the generation of a sincpulse shape as ideal as possible.
The temporal and spectral features of sincshaped pulses bring benefits not only to optical communications but also to many other fields. Actually, sincshaped pulses correspond to the ideal interpolation function for the perfect restoration of bandlimited signals from discrete and noisy data^{23}. Hence, sinc pulses can provide substantial performance improvement to optical sampling devices^{24}. Further, the spectral features of sinc pulses could enable the implementation of ideal rectangular microwave photonics filters^{25,26,27} with tunable passband profiles, thus also providing interesting possibilities for alloptical signal processing^{28}, spectroscopy^{29} and light storage^{30,31}.
Several approaches for the generation of Nyquist pulses have been suggested. In refs 9 and 16, an arbitrary waveform generator was programmed offline to create Nyquist filtering of the baseband signal. This can provide a quite good rolloff factor of β=0.0024 (ref. 16). However, this method is restricted by the speed of electronics because of the limited sampling rate and limited processor capacities, whereas the quality of the Nyquist pulses highly depends on the resolution (number of bits) of digitaltoanalogue converters^{32}. Another possibility is the optical generation of Nyquist pulses^{13,18,20}. These optical sequences can reach much shorter time duration and can thus be multiplexed to an ultrahigh symbol rate. To generate Nyquist pulses, a liquid crystal spatial modulator has been used to shape Gaussian pulses from a modelocked laser into raisedcosine Nyquist pulses. It is also possible to generate Nyquist pulses using fibre optical parametric amplification, pumped by parabolic pulses, and a phase modulator to compensate the pumpinduced chirp^{20}. However, compared with electrical pulse shaping, optical Nyquist pulse generation produces much higher rolloff factors^{33}, such as β=0.5 (refs 13, 18); therefore, multiplexing using this kind of pulses results in a nonoptimal use of bandwidth. Further, most of the reported methods use complex and costly equipment.
In this paper, a method to generate a sequence of very highquality Nyquist pulses with an almost ideal rectangular spectrum (β~0) is proposed and demonstrated. The method is based on the direct synthesization of a flat phaselocked frequency comb with high suppression of outofband components. It is theoretically demonstrated and experimentally confirmed that this comb corresponds to a periodic sequence of timeunlimited sinc pulses. The wide tunability of the method, using a proofofconcept experiment based on two cascaded Mach–Zehnder modulators (MZM), is demonstrated over 4 frequency decades. Experimental results also verify the remarkable high quality of the generated pulses, exhibiting in all cases zero rolloff, minimum spectral broadening when modulated and <1% deviation with respect to the ideal sinc shape. These pulses simultaneously show a minimum ISI and a maximum spectral efficiency, making them an attractive solution for highcapacity TDM–WDM systems.
Results
Basic concepts
Considering that owing to physical limitations the ideal sinc pulse with perfect rectangular optical spectrum has not been demonstrated so far, a different approach for sincshaped Nyquist pulse generation is proposed in this paper. The technique is a straightforward way to realize sincshaped Nyquist pulses in the optical domain, overcoming the limitations imposed by the speed of electronics. The principle of the method is based on the time–frequency duality described by Fourier analysis, as shown in Fig. 2. It is wellknown that a sinc pulse can be represented by a rectangular spectrum in the Fourier domain (see upper figures in Fig. 2), while the frequency content of a train of sinc pulses corresponds to a flat comb with equally spaced components within the bandwidth defined by the singlepulse spectrum (see lower figures in Fig. 2). Therefore, instead of shaping a singlesinc pulse, the approach proposed here produces a sequence of sinc pulses directly from the generation of an optical frequency comb having uniformly spaced components with narrow linewidth, equal amplitude and linearlocked phase, together with strong outerband suppression^{34}. As demonstrated in this paper, the pulse sequence obtained from this rectangular frequency comb is strictly identical to the summation of individual timeunlimited sinc pulses, and intrinsically satisfies the zeroISI Nyquist criterion, similar to the ideal single sincshaped pulse. As described in Fig. 2, the frequency spacing Δf between adjacent spectral lines determines the pulse repetition period T=1/Δf, and the rectangular bandwidth NΔf (N being the number of lines) defines the zerocrossing pulse duration τ_{p}=2/(NΔf). Thus, pulse width and repetition rate can be changed by simply tuning the frequency comb parameters. This feature offers a highly flexible and simple way to adjust the bit rate and bandwidth allocation in an optical network according to actual requirements^{35,36}, or to change the parameters of optical sampling devices^{24} whenever required.
Theory
The Nyquist criterion for a pulse y(t) satisfying zero ISI implies that, for a particular sampling period τ=τ_{p}/2, y(nτ) is 0 for any nonzero integer n, while y(0)≠0. This means that when the signal is periodically sampled with a period τ, a nonzero value is obtained only at the time origin^{11}. For instance, the sinc function defined as is a Nyquist pulse possessing a rectangular spectrum and is therefore unlimited in time. As a consequence of causality, the sinc function is therefore only a theoretical construct^{17}.
In this paper, instead of generating a single timeunlimited sinc pulse, a method to obtain a sequence of sinc pulses is proposed based on the generation of a flat frequency comb with closetoideal rectangular spectrum. Here it is shown that the timedomain representation of the generated comb corresponds to an unlimited ISIfree summation of sincshaped Nyquist pulses. However, in complete contrast to the singlesinc pulse, the pulse sequence can be easily generated from a rectangular frequency comb. Here the mathematical demonstration is presented for an odd number of frequency lines; however, the derivation for an even number can be straightforwardly obtained following the same procedure.
The timedomain representation of the optical field of a frequency comb with N lines, having the same amplitude E_{0}/N and frequency spacing Δf around the central frequency f_{0}, can be expressed as:
For the sake of simplicity, it is assumed that all frequency components have the same phase φ. Strictly speaking, it is sufficient that the phases of all frequency components are locked showing a linear dependence on frequency; however, this linear dependence can be nullified by properly choosing the time origin without the loss of generality. Equal phases will be assumed hereafter to simplify the notation.
From equation (2), the normalized envelope of the optical field is calculated to be , denominated hereafter as periodic sinc function. To demonstrate that this envelope actually corresponds to a train of sincshaped Nyquist pulses, it is convenient to start from its frequency domain representation. According to equation (2) and using the Fourier transform, it follows:
Introducing the rectangular function П that is 1 for all integers n where and 0 elsewhere, the above equation can be written as:
where the rectangular spectrum , covering a bandwidth NΔf, is represented in the time domain by the sincpulse NΔfsinc(NΔft). The temporal dependence of the above expression can then be obtained by taking its inverse Fourier transform and using the Poisson summation formula^{37}:
where ⊗ denotes the convolution operation. Thus, it follows for the righthand side of equation (5):
Therefore, it can be written that
Similarly, for an even number of spectral lines, the envelope of the optical field can be expressed as a train of sinc pulses through the following equation:
where the factor (−1)^{n} comes from the absence of a spectral line at the central frequency of the comb; this eliminates the direct current (DC) component in the optical field envelope.
Comparing equations (7) and (8), the following general expression for the normalized envelope of the optical field resulting from a flat frequency comb is obtained, independent of the parity of N:
The difference in the periodic sinc function for even and odd N can be figured out easily. As depicted in Fig. 3a, all sinc pulses of the pulse train for odd N show the same phase, so that x(t_{s})=1 at every sampling instant for all integer n. For even N, x(t_{s})=(−1)^{n}, so that each pulse envelope is of opposite sign with its preceding and following pulses, as shown in Fig. 3b. Aside from this difference, the optical intensity measured by a photodetector is the same in both cases, and is given by:
Consequently, it is proven that the field envelope of the timedomain representation of a frequency comb of N identical and equally spaced lines corresponds to an infinite summation of sincshaped Nyquist pulses with period and zerocrossing pulse width . Thus, considering that the pulse repetition period is a multiple of the time interval , the resulting timedomain envelope x(t) satisfies the following condition for any integer m:
Thus, the sequence of sinc pulses resulting from a locked phase, rectangular frequency comb satisfies the Nyquist criterion for free ISI within every pulse repetition period T. This condition is automatically and intrinsically satisfied for any flat frequency comb since the number of lines N is an integer by definition. Therefore, the generated sincpulse sequence can be multiplexed in time without ISI.
Proofofconcept experiment
There are several different approaches for the generation of a frequency comb. For instance, they can be obtained from conventional femtosecond lasers, such as Erfibre^{38,39}, Ybfibre^{40} and Ti:sapphire^{41} modelocked lasers, or from a continuous wave optical source exploiting Kerrnonlinearities in an optical resonator^{42,43,44,45}, or employing a combination of strong intensity and phase modulation^{46,47,48} together with chirped Bragg gratings^{49}, dispersive medium^{50} or highly nonlinear fibres^{51,52,53}. However, every comb does not necessarily result in a sequence of Nyquist pulses, since a sincpulse sequence can only be obtained under specific conditions, requiring that the produced comb has to show line amplitudes as equal as possible, linear phase dependence through all lines and a strong suppression of outofband lines. Thus, although flat frequency combs can be obtained using different methods, as for instance through phase modulation^{48,49,50}, the phase difference between lines and the existing outofband components make phase modulators improper for clean generation of sincshaped pulses.
In general, a closetoideal rectangularshaped optical frequency comb can be produced using various implementations; for instance, a nonoptimal frequency comb^{38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53} can be used in combination with a spectral linebyline manipulation of the optical Fourier components^{54,55} to control the amplitude and phase of each spectral line. It turns out that the complexity of this kind of pulse shapers significantly increases with the number of spectral lines, and in general pulse shapers are unable to manipulate a frequency comb having spectrally spaced lines below 1 GHz^{54,55}. Here a simple proofofconcept experimental setup, shown in Fig. 4a, is proposed. This uses two cascaded lithium niobate MZM with a specific adjustment of the bias and modulation voltages (see Methods for details). An optical spectrum analyzer (OSA) with a spectral resolution of 0.01 nm is used to measure the generated frequency combs, while an optical sampling oscilloscope with 500 GHz bandwidth is employed to measure the timedomain pulse train waveforms. While the first modulator, driven by a radiofrequency (RF) signal at a frequency f_{1} is adjusted to generate three seeding spectral components, the second MZM remodulates those lines using an RF signal at f_{2}. Thus, for instance, to generate N=9 spectral lines, the condition f_{1}=3f_{2} or f_{2}=3f_{1} has to be satisfied without any carrier suppression, resulting in a frequency spacing between the lines of Δf=min(f_{1},f_{2}). However, to generate a comb with N=6 lines, the carrier of one of the modulators must be suppressed leading to two possible configurations, as illustrated in Fig. 4b,c. If the optical carrier is suppressed in the first modulator (see Fig. 4b), the RF frequencies must satisfy the condition 2f_{1}=3f_{2}, giving a line spacing Δf=f_{2}. On the other hand, if the carrier is suppressed in the second modulator (see Fig. 4c), the relation between modulating frequencies has to be f_{1}=4f_{2}, resulting in a frequency spacing Δf=2f_{2}.
A highquality rectangularshaped frequency comb can be obtained by tuning the DC bias V_{B} and the RF voltage amplitude ν_{s} of each modulator following the description presented in the Methods section. To ensure that the three components generated by each modulator are in phase, V_{B} and ν_{s} might take either positive or negative values. Moreover, to obtain spectral lines with similar phase using two cascaded MZMs, the phase difference between the modulating RF signals has to be finely adjusted to compensate propagation delays in optical and electrical links, thus leading to almost perfectly shaped symmetric pulses.
On the other hand, to confine the sincpulse sequence into the Nyquist bandwidth, a low modulating voltage ν_{s} must be used to strongly suppress the outofband components. In particular, the RFdriving voltage ν_{s} of both modulators is here adjusted to remain below ~0.36V_{π} (where V_{π} is the halfwave voltage of the MZM), securing a suppression of more than 27 dB for the outofband components. Note that this level of confinement is only possible, thanks to the two degrees of freedom provided by intensity modulators, since both operating bias point and modulating voltages can be adjusted.
Quality and tunability of the sincshaped Nyquist pulses
The quality of the pulses and the flexibility of the method have been experimentally verified by changing the modulating signal frequencies f_{1} and f_{2} in a wide spectral range, and comparing measurements with the theoretical expectations. This way different frequency combs with N=9 spectral components have been generated with a frequency spacing Δf spanning over many decades (between 10 MHz and 10 GHz). In Fig. 5, the measured sinc pulses (black straight lines) are compared with the theoretical ones (reddashed lines) described by equation (10). Measured and theoretical curves are normalized in all figures. Temporal waveforms have been acquired with a sampling interval of 0.2 ps for the case of Δf=10 GHz; this interval has been proportionally increased for longer pulse widths. In particular, Fig. 5a shows the case of modulating frequencies f_{1}=30 MHz and f_{2}=Δf=10 MHz, resulting in sincshaped Nyquist pulses with zerocrossing pulse duration of τ_{p}=22.22 ns, fullwidth at halfmaximum (FWHM) duration of 9.8 ns and a repetition period of T=100 ns. In Fig. 5b–d, the modulating frequencies have been sequentially increased by one order of magnitude. It is observed that the generated pulse sequences coincide very well with the ideal ones over 4 frequency decades, showing a root mean square (r.m.s.) error below 1% for all cases. In addition, it was verified that the spectrum for all these conditions resulted to be close to the ideal rectangular case, as it will be detailed below.
The shaded box in Fig. 6a shows an ideal rectangular spectrum, which corresponds to a singlesinc pulse with a FWHM duration of 9.8 ps, as the one reported in Fig. 5d. The red curve represents the measured flat phaselocked comb in such a case, showing more than 27 dB suppression of the higherorder sidebands and a power difference between components lower than 0.2 dB. The pulse repetition period, corresponding to T=100 ps, is clearly observed in Fig. 6b.
Then, the pulse duration and the repetition rate have been easily changed by modifying the spectral characteristics of the generated frequency comb. For instance, if the second modulator is driven by two RF signals combined in the electrical domain, each of the three frequency components resulting from the first MZM are modulated to create up to five spectral lines each (four sidebands and carrier). This way, N=10 spectral lines separated by Δf=10 GHz have been generated by modulating the first MZM at f_{1}=25 GHz in carrier suppression mode and by driving the second MZM with two RF signals at f_{21}=10 GHz and f_{22}=20 GHz. The measured optical spectrum, showing a bandwidth of 100 GHz and spurious components suppressed by more than 26 dB, is illustrated in Fig. 6c. Note that in this case, the first modulator is working in carrier suppression mode and, therefore, the main spurious lines observed in the spectrum result predominantly from the limited extinction ratio of the modulators (in this case, 40 GHz MZMs with typical extinction ratio of about 23–25 dB), which makes a perfect carrier suppression impossible. Higher suppression of such spurious components can be obtained using modulators with better extinction ratio (note that MZMs with 40 dB extinction ratio are commercially available at 10 GHz bandwidth). Since the frequency spacing among components is the same as in the previous case, that is, Δf=10 GHz, the pulse repetition period T=100 ps has not changed; however, the zerocrossing pulse duration has been reduced down to τ_{p}=20 ps (FWHM duration of 8.9 ps), as shown in Fig. 6d.
By rearranging the modulating frequencies to f_{1}=30, f_{21}=6 and f_{22}=12 GHz, and by adjusting the bias point of the first modulator (see equation (14) in the Methods), so that the carrier is not suppressed in this case, a frequency comb expanding over a bandwidth of 90 GHz has been obtained, with N=15 spectral components, a frequency spacing Δf=6 GHz and more than 27 dB suppression of higherorder sidebands, as reported in Fig. 6e. The measured sinc pulse has a zerocrossing duration of τ_{p}=22 ps (FWHM duration of 9.8 ps) and a repetition period of T=166.67 ps, as depicted in Fig. 6f.
Finally, the bandwidth of the comb has been broadened exploiting the secondorder sidebands of the modulators. As described in the Method section, this can be achieved by using a proper DC bias voltage that suppresses simultaneously all oddorder sidebands; but it also requires a modulating amplitude of ν_{s}≈1.52 V_{π} for a complete carrier suppression. For the MZMs used here, this optimal modulating amplitude corresponds to an RF power of about 1 W. Using standard drivers, it was not possible to reach such an RF power level and suppress completely the carrier, although a strong suppression of unwanted sidebands could be reached by a simple DC bias adjustment. As a workaround, two narrowband fibre Bragg gratings (3 GHz bandwidth each), centred at the carrier wavelength, have been placed at the output of the first MZM, providing more than 40 dB carrier rejection (an optical isolator has also been inserted between the fibre Bragg gratings to avoid multiple reflexions). Thus, driving the first MZM at f_{1}=19.5 GHz, two frequency components (secondorder sidebands) are obtained with a spectral separation of 78 GHz. Then, the second MZM is driven at f_{2}=26 GHz to obtain a comb expanding over a bandwidth of 156 GHz, with N=6 spectral components equally spaced by Δf=26 GHz. The obtained comb is shown in Fig. 6g, presenting a 21dB suppression of unwanted components. In the time domain, the measured sinc pulse has a zerocrossing duration of τ_{p}=12.8 ps (FWHM duration of 5.75 ps) and a repetition period of T=38.46 ps, as shown in Fig. 6h.
Note that the apparent line broadening shown for all frequency components in Fig. 6 results from the limited resolution of the OSA, which is 0.01 nm. The real linewidth is essentially given by the laser linewidth, which is in the kHz range for the used external cavity laser, that is, more than seven orders of magnitude lower than the pulse rectangular bandwidth.
Figure 7a shows a colourgrade plot of the measured Nyquist pulses for the case reported in Fig. 6a,b, demonstrating that even the simple setup proposed in Fig. 4 can generate very stable and highquality sincshaped pulse sequences with very low jitter (82 fs, equivalent to 0.82% of the FWHM) and very high signaltonoise ratio (SNR>40 dB, above the oscilloscope SNR measurement capacity). Jitter and SNR for all other measured conditions exhibit similar values with respect to the ones reported here. The quality of the measured pulses is also analyzed by comparing them with the intensity derived from the analytical expression for Nyquist pulses as a function of the rolloff factor β, as described in equation (1). Figure 7b shows the r.m.s. error between the measured pulses and the theoretical intensity waveforms for rolloff factors between 0 and 1. It can be observed that the minimum r.m.s. error is reached with a factor β=0, indicating that the obtained pulses coincide very well with the ideal sincpulse shape with an r.m.s. error of 0.98%. All other measurements reported in Figs 5 and 6 also present the same quality as the one described here. When this factor β=0 is compared with the rolloff obtained by other optical pulseshaping methods^{13,18,20,21} (reporting β=0.4 in the best case^{21}), a significant improvement in the quality of the pulses generated here can be easily concluded. This is also evident by simply comparing the spectral and timedomain measurements shown in Figs 5 and 6 with results reported in refs 13, 18, 20 and 21.
Discussion
In conclusion, a simple technique to produce sincshaped Nyquist pulses of unprecedented high quality has been proposed and demonstrated based on the optical generation of a phaselocked frequency comb with a rectangular spectral shape. The method offers a high flexibility to modify the pulse parameters, thanks to the possibility of easily changing the bandwidth of the comb, the number of spectral lines and their frequency separation. Because of its conceptual simplicity, many experimental variants can be implemented using similar approaches.
In the context of telecommunication systems, the generated sequence of sincshaped pulses can be multiplexed either in the time or frequency domain following the standard approaches for orthogonal TDM^{18} or Nyquist WDM^{8} transmission schemes. To implement an almost ideal Nyquist transmission system, the zeroISI criterion has to be satisfied by the modulated channels as well. However, it is important to mention that the nearly ideal rectangular spectra reported in Fig. 6 will no longer be obtained if pulses are modulated with data. Since a modulation in time domain corresponds to a convolution in the frequency domain, the spectrum of the modulated sincshaped pulses is given by the convolution of the frequency comb and the frequency representation of the modulating signal. Assuming an ideal rectangular modulation window equal to the pulse repetition period T=1/Δf, the frequency comb will be convolved with a sinc function in the frequency domain^{9} having zero crossings at n·1/T=n·Δf, with n being a nonzero integer number. Thus, the frequency components of the comb coincide with the zero crossings of the modulating signal, which also holds for neighbouring WDM channels (assuming zero guard band).
Figure 8a and c shows the simulated spectra resulting from modulating ideal sinc pulses with onoff keying and binary phaseshift keying modulation formats, respectively. It is possible to observe the expected spectral broadening resulting from the modulation. As can be seen from the dashed lines, the spectral zero crossings outside the Nyquist bandwidth fall exactly in the comb lines of the adjacent WDM channels, indicating that no guard band between the channels is necessary. Thus, this results in an optimal exploitation of the bandwidth. Both simulated conditions have been experimentally verified by modulating the generated sequence of sincshaped pulses using a pseudorandom binary sequence with a length of 2^{31}–1. Figure 8b and d compares the spectral measurements (for onoff keying and binary phaseshift keying modulation, respectively) with the spectrum resulting from the simulations convolved with the spectral response of the OSA (a resolution filter with 0.01 nm bandwidth). It is clearly observed that when the generated sinc pulses are modulated, the spectral broadening matches very well the expected behaviour described by the simulations. The small differences between simulation and experiment come from the nonideal rectangular modulation window and additional convolutions between the very small outofband comb lines and the modulation spectrum.
Measurements and simulations indicate that a spectral broadening, socalled excess bandwidth^{17}, of about 11% results from modulating the generated sinc pulses (considering only the power within the main spectral lobe, confining about 99% of the power). However, different from other optical pulseshaping techniques^{13,18,21}, it is important to notice that this excess bandwidth, expressed as a percentage of the Nyquist frequency, does not depend on the rolloff factor of the unmodulated pulses, since this factor is practically zero in the present case. Instead, the broadening here is only given by the ratio between the pulse repetition rate (defining the modulating window) and the pulse width (defining the Nyquist bandwidth)^{9,17}, thus being proportional to Δf/(NΔf)=1/N (where N is the number of lines in the comb). It is therefore remarkable that even with only N=9 spectral lines, the excess bandwidth resulting from modulation, equal to 1/N=0.11 and here obtained with a simple proofofconcept setup, is significantly lower than the one obtained by other optical pulseshaping methods^{13,18,21}. Such methods actually report a rolloff factor between β=0.4 (ref. 21) and β=0.5 (refs 13, 18) for unmodulated pulses, which is already higher than the factor 0.11 obtained here after modulation. In addition, because of the fixed relation between the symbol duration of the modulating data and the pulse width, this broadening does not require a guard band between WDM channels, as already discussed.
It is worth mentioning that the spectral broadening obtained here can be significantly reduced if the number of lines in the frequency comb is increased^{9,32}. This results in an extension of the modulating window (that is, a narrower modulating spectrum) and/or in a broadening of the Nyquist bandwidth. Thus, for instance, if the pulses in Fig. 6f are modulated, the excess bandwidth would be reduced down to 6.7%. This way, and because of the zero rolloff of the unmodulated pulses, the spectrum of the modulated periodic sinc pulses can expectedly get closer to an ideal rectangular shape^{9,32}.
Finally, in a more general context, it is expected that the use of nearly ideal optical sincshaped pulses would not only increase the transmission data rates in existing optical networks but can also provide great benefits for optical spectroscopy, alloptical sampling devices and photonic analoguetodigital converters, among other potential applications.
Methods
Rectangularshaped frequency comb generation
Consider M intensity modulators, so that each of them can generate two or three equalintensity, phaselocked main spectral lines by controlling its DC bias voltage and RF signal amplitude. The impact of the higherorder sidebands will be addressed in a second stage. If a subset of m modulators each creates three spectral lines (carrier and two firstorder sidebands) and the remaining M−m modulators each produces two lines (two firstorder sidebands with suppressed carrier), a comb with N=2^{M−m}3^{m} equally spaced spectral lines, with the same amplitude and phase, can be generated by cascading the modulators and by properly adjusting the applied bias voltage and modulating amplitude, and by appropriately selecting their modulation frequency.
To properly adjust the DC bias and modulating RF voltage in each MZM, the expression for the output field from each modulator has to be analyzed. If the DC bias and the RF signal voltages applied to a single modulator are V_{B} and ν_{s }cos(ω_{s}t), respectively, its normalized output optical field is given by the expression^{48,56}:
where J_{k} is the Bessel function of the first kind and order k, =V_{B}/V_{π}, and α=v_{s}/V_{π}, in which V_{π} is the halfwave voltage of the modulator. Note that according to equation (12), the amplitude of the carrier, firstorder sidebands and higherorder sidebands can be adjusted by a proper tuning of the RFdriving voltage α and the DC bias ∈. The primary objective is to equalize the amplitudes of the carrier and firstorder sidebands, and the condition to realize it can be found out from the expression of the output field reduced to these three spectral components:
It is important to notice that by using intensity modulators, two degrees of freedom, that is, bias voltage V_{B} and modulating amplitude v_{s}, can be used to equalize the amplitude of the spectral lines having a linear lockedphase difference and to achieve a simultaneous suppression of the higherorder sidebands. This issue makes a significant difference with respect to the use of phase modulators^{46,47,48,49,50,51} where only the modulating voltage can be adjusted, making it impossible to obtain spectral components with the same amplitude and uniformlocked phase.
Figure 9a shows a contour plot representing the amplitude difference between the firstorder sidebands and the carrier (that is, −J_{1}(πα/2)sin(π∈/2)−J_{0}(πα/2)cos(π∈/2)) as a function of the normalized voltages α and ∈. The figure indicates that there are many combinations of α and (represented by the thick solid lines at zero level in the contour plot) that equalize the amplitudes of the carrier and the firstorder sidebands. Actually, as depicted in Fig. 9a, the relation between the optimum bias voltage V_{B} and the driving RF signal amplitude v_{s} that fulfils this condition is a periodic function, which can be simply obtained from equation (13):
Although all valid combinations of V_{B} and v_{s} given by equation (14) and graphed in Fig. 9a provide equalized amplitudes for the three frequency components (two firstorder sidebands and carrier), their absolute amplitude can vary considerably. Moreover, phase and amplitude of the higherorder sidebands can also be adjusted by changing the operating bias point and the modulating RF voltage amplitude. Figure 9b shows the amplitude of the three lowerorder sidebands as a function of the normalized RFdriving voltage α, when the optimum bias is set according to equation (14). It can be observed that a high amplitude of the firstorder sidebands (equal to the carrier amplitude) together with a low amplitude of higherorder sidebands is only possible if the normalized RF voltage α is set to be lower than 0.8. Other voltage conditions result in lower suppression of the higherorder sidebands, leading to a frequency comb with badly equalized frequency components.
It must be pointed out that the higherorder sidebands have to be strongly suppressed to confine the sincpulse sequence into the Nyquist bandwidth. Figure 9c shows the power level of the three higherorder sidebands in dB scale versus the normalized RF voltage, when the bias point is set at its optimum value according to equation (14) (power levels in the figure have been normalized to the maximum power of the equalized firstorder sidebands). The figure points out that, as previously mentioned, strong suppression of the higherorder sidebands can only be achieved by using a low RF signal amplitude. Although only the three lowerorder sidebands are analyzed here, higherorder sidebands are expected to have much reduced power levels because of the lower amplitude of the higherorder Bessel functions J_{k} in this driving voltage range. This can be readily justified as a result from the asymptotic form of the Bessel function J_{k}(x)~x^{k} for small argument x.
According to Fig. 9c, the maximum power of the carrier and the firstorder sidebands can be reached using a driving voltage v_{s}=0.8V_{π}. This condition offers a 15dB suppression of the secondorder sidebands (see reddashed line in the figure). However, stronger higherorder sideband suppression can be achieved by a slight reduction of the driving voltage, which also leads to a small power reduction of carrier and firstorder sidebands. Thus, for instance, using a modulating voltage v_{s}=0.32V_{π}, a higherorder sideband suppression of more than 30 dB can be achieved with a power reduction of 4.5 dB on the carrier and the firstorder sidebands with respect to the maximum reachable power level. Thus, arbitrary outofband suppression can be obtained using lower RF voltages, while the power reduction of carrier and firstorder sidebands can be easily compensated by optical amplification.
To implement the proposed idea, a proofofconcept setup is implemented in this paper based on two cascaded MZM, driven by independent RF generators; however, there are many ways to extend and improve the proposed setup. Instead of a second generator, a frequency tripler and a phase shifter can be used to drive both modulators. In addition, the number of frequency lines generated by each modulator can be increased combining two or even more RF signals in the electrical domain. In this way, the setup can even be compacted to operate using a single MZM.
Further, shorter pulses can be generated with higher bandwidth modulators, or by the exploitation of the secondorder sidebands^{48,56} and the simultaneous suppression of the outofphase components. According to equation (12), all oddorder sidebands can be simultaneously suppressed using a bias voltage V_{B}=∈V_{π}, ∈ being an even number. Under this condition, only the carrier and evenorder sidebands could exit the modulator. While higherorder sidebands are expected to be very low, a strong carrier can still exist. Unfortunately, the carrier component is outofphase with respect to the secondorder sidebands, and therefore it needs to be conveniently suppressed. This suppression can be achieved with a proper RFmodulating amplitude, so that the Bessel function of zero order in equation (12) vanishes. This optimal condition is given by a driving voltage v_{s}≈1.52V_{π}. Figure 9c points out that in such an optimal operating point, the secondorder sidebands can be exploited together with a high suppression of the carrier and oddorder sidebands. This would lead to a broader frequency comb and hence to shorter sincshaped Nyquist pulses. The main practical limitation for this scheme is given by the possibility that the required driving voltage can exceed the maximum RF power allowed by the MZM, and therefore modulators with reduced V_{π} could be more suitable for this purpose.
The proposed technique can produce sincshaped Nyquist pulse sequences of very high quality; however, slight deviations from the ideal sinc shape can be expected in the implementation because of some practical limitations, such as the laser linewidth or the chirp induced by the modulators, leading to small phase differences among the comb spectral components. Possible improvements can be obtained using narrower linewidth optical sources, such as Brillouin lasers with linewidth in the Hz range^{57}, or employing optimized xcut chirpfree intensity modulators^{58}.
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How to cite this article: Soto, M. A. et al. Optical sincshaped Nyquist pulses of exceptional quality. Nat. Commun. 4:2898 doi: 10.1038/ncomms3898 (2013).
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Acknowledgements
T.S. acknowledges the financial support from the EPFL for his visit as a guest professor.
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T.S. and A.V. developed the presented basic idea for the generation of Nyquist pulses. M.A.So. proposed the proofofconcept setup, and M.A. carried out the theoretical analysis and mathematical proofs. M.A.So., M.A., M.A.Sh., A.V. and T.S. contributed to the experiments. C.S.B and L.T supervised the experiments in the Photonic Systems Laboratory and in the Group for Fibre Optics, respectively. All authors contributed to the writing of the manuscript.
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Soto, M., Alem, M., Amin Shoaie, M. et al. Optical sincshaped Nyquist pulses of exceptional quality. Nat Commun 4, 2898 (2013). https://doi.org/10.1038/ncomms3898
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