Optical sinc-shaped Nyquist pulses of exceptional

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Received 27 May 2013 | Accepted 7 Nov 2013 | Published 4 Dec 2013

Marcelo A. Soto1, Mehdi Alem1, Mohammad Amin Shoaie2, Armand Vedadi2, Camille-Sophie Brs2, Luc Thvenaz1 & Thomas Schneider1,w

Sinc-shaped 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 inter-symbol 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 pulse-shaping methods reported so far have remained rather complex and none has led to ideal sinc pulses. Here a method to produce sinc-shaped Nyquist pulses of very high quality is proposed based on the direct synthesis of a rectangular-shaped and phase-locked frequency comb. The method is highly exible 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 sinc-shaped pulses can also bring benets to many other elds, such as microwave photonics, light storage and all-optical sampling.

DOI: 10.1038/ncomms3898 OPEN

Optical sinc-shaped Nyquist pulses of exceptional quality

1 EPFL Swiss Federal Institute of Technology, Group for Fibre Optics, SCI-STI-LT, Station 11, CH-1015 Lausanne, Switzerland. 2 EPFL Swiss Federal Institute of Technology, Photonic Systems Laboratory, STI-IEL-PHOSL, Station 11, CH-1015 Lausanne, Switzerland. w Present address: Institut fr Hochfrequenztechnik,

Hochschule fr Telekommunikation Leipzig, Gustav-Freytag-Stra%e 43-45, 04277 Leipzig, Germany. Correspondence and requests for materials should be addressed to M.A.So. (email: mailto:[email protected]

Web End [email protected] ).

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In currently deployed optical networks, wavelength division multiplexing (WDM) is used to enhance the carrier capacity of optical bres. However, since the data rate in optical networks

increases by close to 29% per year1, new approaches are being developed2. The bulk of these approaches consists in increasing the spectral efciency of optical links. Using multilevel modulation formats and polarization multiplexing, the spectral efciency can be increased from 0.8 to several bit s 1Hz 1 (refs 35). 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 efciency has been envisaged for the next decade6. 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 circuits7. A possible solution is the combination of several lower-rate channels with high spectral efciency into a Tbit s 1 superchannel, which can be routed through the existing optical networks as a single entity8. Such an aggregation can be achieved in the frequency or time domain9. In orthogonal frequency-division multiplexing (OFDM), a superchannel consisting of a set of subcarriers is generated. Each subcarrier exhibits a sinc-shaped spectrum and can therefore be spaced at the baud rate without inter-channel interference. With OFDM, a data rate of26 Tbit s 1 and a net spectral efciency of 5 bit s 1 Hz 1 have been demonstrated10. Similarly, for Nyquist transmission, the symbols are carried by Nyquist pulses11 that overlap in the time domain without ISI. Recently, a 32.5-Tbit s 1 Nyquist WDM transmission with a net spectral efciency of 6.4 bit s 1 Hz 1 has been shown12. Compared with OFDM, Nyquist pulse shaping has several unique advantages as it reduces the receiver complexity13,14, is less sensitive to bre nonlinearities14, requires much lower receiver bandwidths15 and leads to lower peak-to-average power ratios16.

A general expression in the time domain for the amplitude waveform of Nyquist pulses is17,18:

r t

sin 2pt

[afii9848]

2[afii9848]

Time Time

(N-1)[afii9848]

MN1

Figure 1 | Possible multiplexing of sinc-shaped Nyquist pulses. Periodic sinc-pulse sequences can be split into N branches, each of which corresponds to an independent channel. In the nth branch, the periodic sequence is delayed by n times the interval t 1/(NDf), with n 0,...,N 1.

Each channel can be modulated independently with a modulator M0,...,MN 1. These devices can apply any modulation formats to the signal.

Then, the N modulated channels are multiplexed. The shown multiplexing is carried out in the time domain at one carrier wavelength. Since the multiplexed channel shows a sharp-edged spectrum, the next wavelength channel can be directly adjacent to the previous with almost no guard band and can be multiplexed in the time domain in the same way, reaching high temporal and spectral densities together.

2pt

cos 2bpt

; 1

where tp is the pulse duration between zero crossings and b is known as a roll-off factor17, which is in the range 0rbr1.

Among the class of Nyquist pulses11, the sinc-shaped pulse is of particular interest owing to its rectangular spectrum17 and zero roll-off. This allows minimizing the guard band between optical channels. Theoretically, for a sinc-pulse Nyquist transmission, each symbol consists of a time-unlimited sinc-pulse. However, since causality makes it impossible, periodic pulses are typically used in every experimental demonstration of Nyquist pulse transmission1220. 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 time-division 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 time-domain 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, time-domain channels can be multiplexed at different wavelengths; this is designated as Nyquist WDM8 where pulses can be generated and modulated for each carrier. Since higher-order modulation formats, multiplexing, transmission and

demultiplexing of Nyquist pulses have already been shown elsewhere1218, here the focus is placed on the generation of a sinc-pulse shape as ideal as possible.

The temporal and spectral features of sinc-shaped pulses bring benets not only to optical communications but also to many other elds. Actually, sinc-shaped pulses correspond to the ideal interpolation function for the perfect restoration of band-limited signals from discrete and noisy data23. Hence, sinc pulses can provide substantial performance improvement to optical sampling devices24. Further, the spectral features of sinc pulses could enable the implementation of ideal rectangular microwave photonics lters2527 with tunable passband proles, thus also providing interesting possibilities for all-optical signal processing28, spectroscopy29 and light storage30,31.

Several approaches for the generation of Nyquist pulses have been suggested. In refs 9 and 16, an arbitrary waveform generator was programmed ofine to create Nyquist ltering of the baseband signal. This can provide a quite good roll-off factor of b 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 digital-to-analogue converters32. Another possibility is the optical generation of Nyquist pulses13,18,20. 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 mode-locked laser into raised-cosine Nyquist pulses. It is also possible to generate Nyquist pulses using bre optical parametric amplication, pumped by parabolic pulses, and a phase modulator to compensate the pump-induced chirp20. However, compared with electrical pulse shaping, optical Nyquist pulse generation produces much higher roll-off factors33, such as b 0.5 (refs 13,18); therefore, multiplexing using this kind of

pulses results in a non-optimal 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 high-quality Nyquist pulses with an almost ideal rectangular spectrum (bB0) is proposed and demonstrated. The method is based on the direct synthesization of a at phase-locked frequency comb with high suppression of out-of-band components. It is theoretically demonstrated and experimentally conrmed that this comb corresponds to a periodic sequence of time-unlimited

4bt

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sinc pulses. The wide tunability of the method, using a proof-of-concept experiment based on two cascaded MachZehnder 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 roll-off, minimum spectral broadening when modulated and o1% deviation with respect to the ideal sinc shape. These pulses simultaneously show a minimum ISI and a maximum spectral efciency, making them an attractive solution for high-capacity TDMWDM systems.

ResultsBasic 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 sinc-shaped Nyquist pulse generation is proposed in this paper. The technique is a straightforward way to realize sinc-shaped Nyquist pulses in the optical domain, overcoming the limitations imposed by the speed of electronics. The principle of the method is based on the timefrequency duality described by Fourier analysis, as shown in Fig. 2. It is well-known that a sinc pulse can be represented by a rectangular spectrum in the Fourier domain (see upper gures in Fig. 2), while the frequency content of a train of sinc pulses corresponds to a at comb with equally spaced components within the bandwidth dened by the single-pulse spectrum (see lower gures in Fig. 2). Therefore, instead of shaping a single-sinc 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 linear-locked phase, together with strong outer-band suppression34. As demonstrated in this paper, the pulse sequence obtained from this rectangular frequency comb is strictly identical to the summation of individual time-unlimited sinc pulses, and intrinsically satises the zero-ISI Nyquist criterion, similar to the ideal single sinc-

shaped pulse. As described in Fig. 2, the frequency spacing Df between adjacent spectral lines determines the pulse repetition period T 1/Df, and the rectangular bandwidth NDf (N being the

number of lines) denes the zero-crossing pulse duration tp

2/(NDf). Thus, pulse width and repetition rate can be changed by simply tuning the frequency comb parameters. This feature offers a highly exible and simple way to adjust the bit rate and bandwidth allocation in an optical network according to actual requirements35,36, or to change the parameters of optical sampling devices24 whenever required.

Theory. The Nyquist criterion for a pulse y(t) satisfying zero ISI implies that, for a particular sampling period t tp/2, y(nt) is 0

for any non-zero integer n, while y(0)a0. This means that when the signal is periodically sampled with a period t, a non-zero value is obtained only at the time origin11. For instance, the sinc function dened as sinct sinptpt 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 construct17.

In this paper, instead of generating a single time-unlimited sinc pulse, a method to obtain a sequence of sinc pulses is proposed based on the generation of a at frequency comb with close-to-ideal rectangular spectrum. Here it is shown that the time-domain representation of the generated comb corresponds to an unlimited ISI-free summation of sinc-shaped Nyquist pulses. However, in complete contrast to the single-sinc 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 time-domain representation of the optical eld of a frequency comb with N lines, having the same amplitude E0/N

and frequency spacing Df around the central frequency f0, can be expressed as:

E t

a b

Time domain

Frequency domain

N 1

n N 12

2 e2ip f0 nDft if

N e2ipf0t if

N 1

n N 12

2 e2ipnDft

[afii9848]p = 2/(Nf )

Time Frequency

N e2ipf0t if

eipDfN 1t e ipDfN 1t e2ipDft 1

E0 sin pNDft

c d

T = 1/f

Nsin pDft

Figure 2 | Timefrequency correspondence for sinc-shaped Nyquist pulses. Time (left) and frequency (right) representation of a single-sinc pulse (top) and a sinc-pulse sequence (bottom). Since the directly observed quantity in the optical domain is proportional to the optical intensity (or power), here the gure shows the intensity of the time-domain traces instead of the eld amplitude. The Fourier domain representation of a sinc pulse (a) is a rectangular function (b), while the spectrum of an unlimited sinc-pulse sequence (c) is a frequency comb with uniform phase under a rectangular envelope (d).

e2ipf0t if: 2

For the sake of simplicity, it is assumed that all frequency components have the same phase f. Strictly speaking, it is sufcient that the phases of all frequency components are locked showing a linear dependence on frequency; however, this linear dependence can be nullied 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 eld is calculated to be sinpNDftNsinpDft, denominated hereafter as periodic sinc function. To demonstrate that this envelope actually corresponds to a train of sinc-shaped Nyquist pulses, it is convenient to start from its frequency domain representation. According to equation (2) and using the Fourier transform, it

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follows:

F sin pNDft

1.5

sin pDft

2 e2ipnDft 8<

9 =

;

N 1

n N 12

N 1

n N 12

2 d f nDf

Normalized amplitude

0.5

Introducing the rectangular function & n

that is 1 for all

0.5

integers n where n

j j N 12 and 0 elsewhere, the above equation

can be written as:

N 1

n N 12

1.5 0 1 2 3 4 5

Time (t/T)

2 d f nDf

X 1n 1 n N

d f nDf

f NDf

1.5

1 n 1

d f nDf

;

Normalized amplitude

0.5

where the rectangular spectrum f

NDf

, covering a bandwidth NDf, is represented in the time domain by the sinc-pulse

NDf sinc(NDft). The temporal dependence of the above expression can then be obtained by taking its inverse Fourier transform and using the Poisson summation formula37:

F 1 f NDf

n 1

0.5

1.5 0 1 2 3 4 5

d f nDf

( )

Nsinc NDft

Time (t/T)

where # denotes the convolution operation. Thus, it follows for the right-hand side of equation (5):

Nsinc NDft

X 1n 1d t nn Df

1 n 1

X 1n 1d t n Df

;

Figure 3 | Normalized eld envelope of a frequency comb. (a) Odd (N 9) and (b) even (N 8) number of spectral lines. The time axis is

normalized with respect to the pulse period T. An odd number of lines leads to a sequence of in-phase sinc-shaped Nyquist pulses, while an even number N results in a sequence with alternated p-phase-modulated pulses.

Nsinc NDf t

n Df

Therefore, it can be written that

sin pNDft

Nsin pDft

1 n 1

: 10

Consequently, it is proven that the eld envelope of the time-domain representation of a frequency comb of N identical and equally spaced lines corresponds to an innite summation of sinc-shaped Nyquist pulses with period 1Df and zero-crossing pulse width 2

NDf . Thus, considering that the pulse repetition period T 1Df is a multiple of the time interval t

sinc NDf t

n Df

: 7

Similarly, for an even number of spectral lines, the envelope of the optical eld can be expressed as a train of sinc pulses through the following equation:

sin pNDft

Nsin pDft

1 n 1

NDf , the resulting time-domain envelope x(t) satises the following condition for any integer m:

x mt

n Df

; 8

N 1m=N m:::; 2N; N; 0; N; 2N; :::

0 otherwise:

nsinc NDf t

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 eld envelope.

Comparing equations (7) and (8), the following general expression for the normalized envelope of the optical eld resulting from a at frequency comb is obtained, independent of the parity of N:

sin pNDft

Nsin pDft

1 n 1

N 1nsinc NDf t

n Df

: 9

Thus, the sequence of sinc pulses resulting from a locked phase, rectangular frequency comb satises the Nyquist criterion for free ISI within every pulse repetition period T. This condition is automatically and intrinsically satised for any at frequency comb since the number of lines N is an integer by denition. Therefore, the generated sinc-pulse sequence can be multiplexed in time without ISI.

Proof-of-concept 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 Er-bre38,39, Yb-bre40 and Ti:sapphire41 mode-locked lasers, or from a continuous wave optical source exploiting Kerr-nonlinearities in an optical resonator4245, or employing a combination of strong intensity and phase modulation4648 together with chirped Bragg gratings49, dispersive medium50 or highly nonlinear bres5153. However, every comb does not necessarily result in a sequence of Nyquist pulses, since a sinc-

sin pNDftNsinpDft for

even and odd N can be gured out easily. As depicted in Fig. 3a, all sinc pulses of the pulse train for odd N show the same phase, so that x(ts) 1 at every sampling instant ts nDf for all integer n.

For even N, x(ts) ( 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:

I t

E t

j j2E20

sin2pNDft N2sin2pDft

The difference in the periodic sinc function x t

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pulse sequence can only be obtained under specic 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 out-of-band lines. Thus, although at frequency combs can be obtained using different methods, as for instance through phase modulation4850, the phase difference between lines and the existing out-of-band components make phase modulators improper for clean generation of sinc-shaped pulses.

In general, a close-to-ideal rectangular-shaped optical frequency comb can be produced using various implementations; for instance, a non-optimal frequency comb3853 can be used in combination with a spectral line-by-line manipulation of the optical Fourier components54,55 to control the amplitude and phase of each spectral line. It turns out that the complexity of this kind of pulse shapers signicantly 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 GHz54,55. Here a simple proof-of-concept experimental set-up, shown in Fig. 4a, is proposed. This uses two cascaded lithium niobate MZM with a specic 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 time-domain pulse train waveforms. While the rst modulator, driven by a radio-frequency (RF) signal at a frequency f1 is adjusted to generate three seeding spectral components, the second MZM re-modulates those lines using an RF signal at f2. Thus, for instance, to generate N 9

spectral lines, the condition f1 3f2 or f2 3f1 has to be satised

without any carrier suppression, resulting in a frequency spacing between the lines of Df min(f1,f2). However, to generate a comb

with N 6 lines, the carrier of one of the modulators must be

suppressed leading to two possible congurations, as illustrated in Fig. 4b,c. If the optical carrier is suppressed in the rst modulator (see Fig. 4b), the RF frequencies must satisfy the condition 2f1 3f2, giving a line spacing Df f2. On the other hand, if the

carrier is suppressed in the second modulator (see Fig. 4c), the relation between modulating frequencies has to be f1 4f2,

resulting in a frequency spacing Df 2f2.

A high-quality rectangular-shaped frequency comb can be obtained by tuning the DC bias VB and the RF voltage amplitude ns of each modulator following the description presented in the Methods section. To ensure that the three components generated by each modulator are in phase, VB and ns 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 nely adjusted to compensate propagation delays in optical and electrical links, thus leading to almost perfectly shaped symmetric pulses.

On the other hand, to conne the sinc-pulse sequence into the Nyquist bandwidth, a low modulating voltage ns must be used to strongly suppress the out-of-band components. In particular, the RF-driving voltage ns of both modulators is here adjusted to remain below B0.36Vp (where Vp is the half-wave voltage of the

MZM), securing a suppression of more than 27 dB for the out-of-band components. Note that this level of connement 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 sinc-shaped Nyquist pulses. The quality of the pulses and the exibility of the method have been experimentally veried by changing the modulating signal

frequencies f1 and f2 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 Df 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 (red-dashed lines) described by equation (10). Measured and theoretical curves are normalized in all gures. Temporal waveforms have been acquired with a sampling interval of 0.2 ps for the case of Df 10 GHz; this interval has been proportionally

increased for longer pulse widths. In particular, Fig. 5a shows the case of modulating frequencies f1 30 MHz and f2 Df 10

MHz, resulting in sinc-shaped Nyquist pulses with zero-crossing pulse duration of tp 22.22 ns, full-width at half-maximum

(FWHM) duration of 9.8 ns and a repetition period of T 100 ns.

In Fig. 5bd, 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 veried 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 single-sinc pulse with a FWHM duration of 9.8 ps, as the one reported in Fig. 5d. The red curve represents the measured at phase-locked comb in such a case, showing more than 27 dB suppression of the higher-order 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.

BIAS BIAS

Synchro

Optical spectrum

analyzer

Optical sampling

oscilloscope

ECL MZM1 MZM2

Phase

RF-generator

b c

f2 f2 f2 f2 f2 f2 f2 f2 f2 f2

f1 f1 f1 f1

Figure 4 | Basic experimental implementation. (a) Proof-of-concept setup. Solid and dashed lines describe optical and electrical connections, respectively. An external cavity laser (ECL) generates a narrow linewidth continuous wave light at 1,550 nm. MZM1 generates spectral lines separated by a frequency f1. Then, MZM2 re-modulates these seeding components with a frequency f2. RF power and DC bias in both MZMs are adjusted so that all lines result with the same amplitude and phase, and additional sidebands are highly suppressed. An undistorted waveform is only obtained with a proper adjustment of the relative modulating phase, and therefore both RF generators have been synchronized using a common time base. (b) Generation of a frequency comb with N 6 lines. MZM1 is

driven with a frequency f1 and operates in carrier suppression mode, so that MZM2 re-modulates the two seeding lines, with no carrier suppression, at a frequency f2 Df. (c) Second option to generate a comb with N 6

spectral lines; in this case, MZM1 is driven with a frequency f1 (no carrier suppression), while MZM2 re-modulates the three seeding components, in carrier suppression mode, at a frequency f2 f1/4 Df/2.

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Normalized intensity (a.u.)

0.8

Repetition period: 100 ns Pulse FWHM: 9.8 ns

Pulse bandwidth: 90 MHz

Pulse bandwidth: 900 MHz

Repetition period: 10 ns Pulse FWHM: 0.98 ns

0.6

0.4

0.2

20 0 20 40 60 80

2 0 2 4 6 8

Time (ns)

Normalized intensity (a.u.)

0.8

Pulse bandwidth: 9 GHz

Repetition period: 1 ns Pulse FWHM: 98 ps

Pulse bandwidth: 90 GHz

Repetition period: 100 ps Pulse FWHM: 9.8 ps

0.6

0.4

0.2

200 0 200 400 600 800

20 0 20 40 60 80

Time (ps)

Figure 5 | Tunability of sinc-shaped Nyquist pulses using nine spectral lines. Sinc-shaped Nyquist pulses measured using a 500-GHz optical sampling oscilloscope. The calculated waveforms (red-dashed lines) according to equation (10) are compared with the measured pulses (black straight lines) for different bandwidth conditions over 4 decades. Nyquist pulses are obtained from the generation of a rectangular frequency comb with nine phase-locked components spanning over a spectral width between 90 MHz and 90 GHz, using modulating frequencies (a) f1 30 MHz and f2 Df 10 MHz,

(b) f1 300 MHz and f2 Df 100 MHz, (c) f1 3 GHz and f2 Df 1 GHz, and (d) f1 30 GHz and f2 Df 10 GHz. The maximum difference between

measured pulses and theoretical ones remained in all cases below 1%.

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 rst MZM are modulated to create up to ve spectral lines each (four sidebands and carrier). This way, N 10 spectral lines separated

by Df 10 GHz have been generated by modulating the rst

MZM at f1 25 GHz in carrier suppression mode and by driving

the second MZM with two RF signals at f21 10 GHz and

f22 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 rst 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 2325 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, Df 10

GHz, the pulse repetition period T 100 ps has not changed;

however, the zero-crossing pulse duration has been reduced down to tp 20 ps (FWHM duration of 8.9 ps), as shown in Fig. 6d.

By rearranging the modulating frequencies to f1 30, f21 6

and f22 12 GHz, and by adjusting the bias point of the rst

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 Df 6 GHz and more than

27 dB suppression of higher-order sidebands, as reported in Fig. 6e. The measured sinc pulse has a zero-crossing duration of

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 second-order 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 odd-order sidebands; but it also requires a modulating amplitude of nsE1.52 Vp 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 bre Bragg gratings (3 GHz bandwidth each), centred at the carrier wavelength, have been placed at the output of the rst MZM, providing more than 40 dB carrier rejection (an optical isolator has also been inserted between the bre Bragg gratings to avoid multiple reexions). Thus, driving the rst MZM at f1 19.5 GHz, two frequency

components (second-order sidebands) are obtained with a spectral separation of 78 GHz. Then, the second MZM is driven at f2 26 GHz to obtain a comb expanding over a bandwidth of

156 GHz, with N 6 spectral components equally spaced by

Df 26 GHz. The obtained comb is shown in Fig. 6g, presenting a

21-dB suppression of unwanted components. In the time domain, the measured sinc pulse has a zero-crossing duration of tp 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.

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90 GHz

27 dB

Optical power (dBm)

Optical power (dBm) Optical power (dBm) Optical power (dBm)

Normalized intensity (a.u.)

100 ps

9.8 ps

0.8

0.6

0.4

0.2

30 1,549.4 1,549.8 1,550.2 1,550.6

150 100 0 50 150

50 100

Wavelength (nm)

Time (ps)

100 GHz

26 dB

Normalized intensity (a.u.)

100 ps

8.9 ps

0.8

0.6

0.4

0.2

30 1,549.4 1,549.8 1,550.2 1,550.6

150 100 0 50 150

50 100

200 100 0 100 200

Wavelength (nm)

Time (ps)

90 GHz

27 dB

166.67 ps

9.8 ps

0.8

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0.6

0.4

0.2

30 1,549.4 1,549.8 1,550.2 1,550.6

Wavelength (nm)

Time (ps)

156 GHz

38.46 ps

5.75 ps

0.8

Normalized intensity (a.u.)

21 dB

0.6

0.4

0.2

40 1,549.4 1,549.8 1,550.2 1,550.6

50 0 50

Wavelength (nm)

Time (ps)

Figure 6 | Frequency and time-domain representation of the generated sinc-shaped Nyquist pulses. Pulse duration and repetition rate can be easily modied by adjusting the bias voltage of the modulators as well as the frequency and amplitude of modulating signals. Frequency combs with different bandwidth and number of spectral components have been experimentally generated. (a) Measured spectrum and (b) measured time-domain waveform of a comb generated with N 9 spectral components separated by Df 10 GHz, and expanding over a bandwidth of 90 GHz. (c) Spectrum and (d) time-

domain waveform of a comb generated with N 10, Df 10 GHz, and bandwidth of 100 GHz. (e) Spectrum and (f) time-domain waveform of a comb

generated with N 15, Df 6 GHz, and bandwidth of 90 GHz. (g) Spectrum and (h) time-domain waveform of a comb with N 6, Df 26 GHz, and an

extended bandwidth of 156 GHz. The comb has been spectrally broadened using the second-order sidebands of the rst MZM. A power difference among spectral components lower than 0.2 dB is obtained in all cases. The shaded boxes in a,c,e and g represent the theoretical Nyquist bandwidth of the generated sinc pulses. Spectral measurements are obtained with a resolution of 0.01 nm, temporal waveforms acquired with a 500-GHz optical oscilloscope using a sampling interval of 0.2 ps and two time-averaged traces. Only the waveform in h is measured with eight times averaging.

Figure 7a shows a colour-grade plot of the measured Nyquist pulses for the case reported in Fig. 6a,b, demonstrating that even the simple set-up proposed in Fig. 4 can generate very stable and high-quality sinc-shaped pulse sequences with very low jitter (82 fs, equivalent to 0.82% of the FWHM) and very high signal-to-noise ratio (SNR440 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 roll-off factor b, as described in equation (1). Figure 7b shows the r.m.s. error between the measured pulses and the theoretical intensity waveforms for roll-off factors between 0 and 1. It can be observed that the minimum r.m.s. error is reached with a factor b 0,

indicating that the obtained pulses coincide very well with the ideal sinc-pulse 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 b 0

is compared with the roll-off obtained by other optical

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SNR >40 dB Jitter: 82 fs

Signal power (mW)

40 35 30 25 20 15 10

5 0

150 100 50 0 50 100 150

Time (ps)

r.m.s. error (%)

Minimum r.m.s. error:0.98%

0 0 0.2 0.4 0.6 0.8 1

Roll-off factor

Figure 7 | High stability and quality of periodic sinc pulses. (a) Colour-grade gure for one of the measured sinc-pulse sequences. In this case, a frequency comb with N 9 spectral components separated by Df 10 GHz

is generated (corresponding to the case depicted in Fig. 6a,b, but with no averaging). Measurement indicates a jitter of 82 fs and a SNR440 dB.

Other generated pulse sequences exhibit it similar levels of jitter and SNR. (b) r.m.s. error between measured pulses and the theoretical Nyquist pulse intensity derived from equation (1) as a function of the roll-off factor b.

The r.m.s. error is minimized for b 0, indicating that the generated pulses

match very well the ideal sinc shape with an error of 0.98%. Waveforms measured with other modulating frequencies exhibit similar behaviour.

pulse-shaping methods13,18,20,21 (reporting b 0.4 in the best

case21), a signicant improvement in the quality of the pulses generated here can be easily concluded. This is also evident by simply comparing the spectral and time-domain measurements shown in Figs 5 and 6 with results reported in refs 13,18,20 and 21.

DiscussionIn conclusion, a simple technique to produce sinc-shaped Nyquist pulses of unprecedented high quality has been proposed and demonstrated based on the optical generation of a phase-locked frequency comb with a rectangular spectral shape. The method offers a high exibility 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 sinc-shaped pulses can be multiplexed either in the time or frequency domain following the standard approaches for orthogonal TDM18 or Nyquist WDM8 transmission schemes. To implement an almost ideal Nyquist transmission system, the zero-ISI criterion has to be satised 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 sinc-shaped 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/Df, the frequency comb will be convolved with a

sinc function in the frequency domain9 having zero crossings at n 1/T n Df, with n being a non-zero 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 on-off keying and binary phase-shift 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 veried by modulating the generated sequence of sinc-shaped pulses using a pseudo-random binary sequence with a length of 2311. Figure 8b and d compares the spectral measurements (for on-off keying and binary phase-shift keying modulation, respectively) with the spectrum resulting from the simulations convolved with the spectral response of the OSA (a resolution lter 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 non-ideal rectangular modulation window and additional convolutions between the very small out-of-band comb lines and the modulation spectrum.

Measurements and simulations indicate that a spectral broadening, so-called excess bandwidth17, of about 11% results from modulating the generated sinc pulses (considering only the power within the main spectral lobe, conning about 99% of the power). However, different from other optical pulse-shaping techniques13,18,21, it is important to notice that this excess bandwidth, expressed as a percentage of the Nyquist frequency, does not depend on the roll-off 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 (dening the modulating window) and the pulse width (dening the Nyquist bandwidth)9,17, thus being proportional to Df/(NDf) 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 proof-of-

concept set-up, is signicantly lower than the one obtained by other optical pulse-shaping methods13,18,21. Such methods actually report a roll-off factor between b 0.4 (ref. 21) and

b 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 xed 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 signicantly reduced if the number of lines in the frequency comb is increased9,32. 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 roll-off of the unmodulated pulses, the spectrum of the modulated periodic sinc pulses can expectedly get closer to an ideal rectangular shape9,32.

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms3898 ARTICLE

Normalized power (dB)

Measurement Simulation

70 60 40 20 0 20 40 60

35 60 40 20 0 20 40 60

Frequency detuning (GHz)

Normalized power (dB)

30 60 40 20 0 20 40 60

Frequency detuning (GHz)

Figure 8 | Spectrum of modulated sinc pulses. (a) Simulated spectrum resulting from modulating ideal sinc-shaped pulses with on-off keying (OOK) format, using an ideal rectangular modulating window. (b) Measured spectrum obtained from the OOK modulation of the generated sequence of sinc-shaped pulses using a pseudo-random binary sequence of length 231 1. The measured spectrum (black straight line) is compared with the simulated

one reported in a convolved with the nite spectral bandwidth (0.01 nm) of the OSA (red-dashed line). (c) Simulated spectrum resulting from modulating ideal sinc-shaped pulses with binary phase-shift keying (BPSK) format using an ideal rectangular modulating window. (d) Measured spectrum obtained from modulating the generated sequence of sinc-shaped pulses with BPSK. The measured spectrum (black straight line) is compared with the simulated one reported in c convolved with the ltering bandwidth of the OSA (red-dashed line). The dotted boxes in a and c show the rectangular spectrum of one single pulse and the dashed lines indicate the position of the two adjacent WDM channels, showing that although the spectrum is broadened by the modulation, no guard band between the channels is necessary.

Finally, in a more general context, it is expected that the use of nearly ideal optical sinc-shaped pulses would not only increase the transmission data rates in existing optical networks but can also provide great benets for optical spectroscopy, all-optical sampling devices and photonic analogue-to-digital converters, among other potential applications.

Methods

Rectangular-shaped frequency comb generation. Consider M intensity modulators, so that each of them can generate two or three equal-intensity, phase-locked main spectral lines by controlling its DC bias voltage and RF signal amplitude. The impact of the higher-order sidebands will be addressed in a second stage. If a subset of m modulators each creates three spectral lines (carrierand two rst-order sidebands) and the remaining M m modulators each pro

duces two lines (two rst-order sidebands with suppressed carrier), a comb with N 2M m3m 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 eld from each modulator has to be analyzed. If the DC bias and the RF signal voltages applied to a single modulator are VB and nscos(ost), respectively, its normalized output optical eld is given by the expression48,56:

E t

X 1k 1 1 k cospE2

J2k pa 2

cos o0t 2kost

J1 pa 2

cos o0 os

f g: 13

It is important to notice that by using intensity modulators, two degrees of freedom, that is, bias voltage VB and modulating amplitude vs, can be used to equalize the amplitude of the spectral lines having a linear locked-phase difference and to achieve a simultaneous suppression of the higher-order sidebands. This issue makes a signicant difference with respect to the use of phase modulators4651 where only the modulating voltage can be adjusted, making it impossible to obtain spectral components with the same amplitude and uniform-locked phase.

Figure 9a shows a contour plot representing the amplitude difference between the rst-order sidebands and the carrier (that is, J1(pa/2)sin(pE/2) J0(pa/2)

cos(pE/2)) as a function of the normalized voltages a and E. The gure indicates that there are many combinations of a and E (represented by the thick solid lines at zero level in the contour plot) that equalize the amplitudes of the carrier and the rst-order sidebands. Actually, as depicted in Fig. 9a, the relation between the optimum bias voltage VB and the driving RF signal amplitude vs that fulls this condition is a periodic function, which can be simply obtained from equation (13):

2Vpp tan 1

8 <

J0 pv

J1 pv

9 =

;

sin pE 2

J2k 1

pa2

cos o0t 2k 1

: 14

Although all valid combinations of VB and vs given by equation (14) and graphed in Fig. 9a provide equalized amplitudes for the three frequency components (two rst-order sidebands and carrier), their absolute amplitude can vary considerably. Moreover, phase and amplitude of the higher-order 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 lower-order sidebands as a function of the normalized RF-driving voltage a, when the optimum bias is set according to equation (14). It can be observed that a high amplitude of the rst-order sidebands (equal to the carrier amplitude) together with a low amplitude of higher-order sidebands is only possible if the normalized RF voltage a is set to be lower than 0.8. Other voltage conditions result in lower suppression of the higher-order sidebands, leading to a frequency comb with badly equalized frequency components.

It must be pointed out that the higher-order sidebands have to be strongly suppressed to conne the sinc-pulse sequence into the Nyquist bandwidth.

ost

g; 12

where Jk is the Bessel function of the rst kind and order k, E VB/Vp, and a

vs/Vp, in which Vp is the half-wave voltage of the modulator. Note that according to equation (12), the amplitude of the carrier, rst-order sidebands and higher-order sidebands can be adjusted by a proper tuning of the RF-driving voltage a and the

DC bias E. The primary objective is to equalize the amplitudes of the carrier and rst-order sidebands, and the condition to realize it can be found out from the

expression of the output eld reduced to these three spectral components:

E t

cos

J0 pa 2

cos o0t sin

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0.4

Normalized bias voltage ( )

0.8

0 0.8

0.4

0 0.4

0.4

0.8

0.5

0.4

0 0.4

0.4

0.8

0.4

0.8

0.4

0.5

0.4

4 4 2 0 2 4

0.4

Normalized driving voltage ()

0.6

Normalized amplitude (a.u.)

0.6 0 1 2 3 4

0.3

1st order and carrier

2nd order

3rd order

0.3

Normalized driving voltage ([afii9825])

1st orderand carrier 2nd order 3rd order

Normalized intensity (dB)

40 0 1 2 3 4

Normalized driving voltage ([afii9825])

Figure 9 | Rectangular frequency comb generation with MZMs. (a) Amplitude difference between rst-order sidebands and carrier component [ J1(pa/2)sin (pE/2) J0(pa/2)cos (pE/2)]. Equalization of the amplitude

between the two rst-order sidebands and carrier is only possible if pairs of bias voltage E and driving voltage a lying over the thick black line at zero level are used. This amplitude equalization not only leads to frequency components with the same power level but also ensures the same phase between them. (b) Field amplitude and (c) power of the three lower-order sidebands as a function of the normalized RF voltage, when the DC bias is set to equalize carrier and rst-order sideband amplitudes. Power levels have been normalized to the maximum power reached by the rst-order sidebands.

Figure 9c shows the power level of the three higher-order 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 gure have been normalized to the maximum power of the equalized rst-order sidebands). The gure points out that, as previously mentioned, strong suppression of the higher-order sidebands can only be achieved by using a low RF signal amplitude. Although only the three lower-order sidebands are analyzed here, higher-order sidebands are expected to have much reduced power levels because of the lower amplitude of the higher-order Bessel functions Jk in this driving voltage range. This can be readily justied as a result from the asymptotic form of the Bessel function Jk(x)Bxk for small

argument x.

According to Fig. 9c, the maximum power of the carrier and the rst-order sidebands can be reached using a driving voltage vs 0.8Vp. This condition offers a

15-dB suppression of the second-order sidebands (see red-dashed line in the gure). However, stronger higher-order sideband suppression can be achieved by a slight reduction of the driving voltage, which also leads to a small power reduction of carrier and rst-order sidebands. Thus, for instance, using a modulating voltage vs 0.32Vp, a higher-order sideband suppression of more than 30 dB can be

achieved with a power reduction of 4.5 dB on the carrier and the rst-order sidebands with respect to the maximum reachable power level. Thus, arbitrary outof-band suppression can be obtained using lower RF voltages, while the power

reduction of carrier and rst-order sidebands can be easily compensated by optical amplication.

To implement the proposed idea, a proof-of-concept set-up 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 set-up. 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 set-up 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 second-order sidebands48,56 and the simultaneous suppression of the out-of-phase components. According to equation (12), all odd-order sidebands can be simultaneously suppressed using a bias voltage VB EVp, E

being an even number. Under this condition, only the carrier and even-order sidebands could exit the modulator. While higher-order sidebands are expected to be very low, a strong carrier can still exist. Unfortunately, the carrier component is out-of-phase with respect to the second-order sidebands, and therefore it needs to be conveniently suppressed. This suppression can be achieved with a proper RF-modulating amplitude, so that the Bessel function of zero order inequation (12) vanishes. This optimal condition is given by a driving voltage vsE1.52Vp. Figure 9c points out that in such an optimal operating point, the second-order sidebands can be exploited together with a high suppression of the carrier and odd-order sidebands. This would lead to a broader frequency comb and hence to shorter sinc-shaped 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 Vp could be more suitable for this purpose.

The proposed technique can produce sinc-shaped 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 range57, or employing optimized x-cut chirp-free intensity modulators58.

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Acknowledgements

T.S. acknowledges the nancial support from the EPFL for his visit as a guest professor.

Author contributions

T.S. and A.V. developed the presented basic idea for the generation of Nyquist pulses. M.A.So. proposed the proof-of-concept set-up, 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.

Additional information

Competing nancial interests: The authors declare no competing nancial interests.

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How to cite this article: Soto, M. A. et al. Optical sinc-shaped Nyquist pulses of exceptional quality. Nat. Commun. 4:2898 doi: 10.1038/ncomms3898 (2013).

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NATURE COMMUNICATIONS | 4:2898 | DOI: 10.1038/ncomms3898 | http://www.nature.com/naturecommunications

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Sinc-shaped 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 inter-symbol 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 pulse-shaping methods reported so far have remained rather complex and none has led to ideal sinc pulses. Here a method to produce sinc-shaped Nyquist pulses of very high quality is proposed based on the direct synthesis of a rectangular-shaped and phase-locked 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 sinc-shaped pulses can also bring benefits to many other fields, such as microwave photonics, light storage and all-optical sampling.

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Title

Optical sinc-shaped Nyquist pulses of exceptional quality

Author

Soto, Marcelo A; Alem, Mehdi; Amin Shoaie, Mohammad; Vedadi, Armand; Brès, Camille-sophie; Thévenaz, Luc; Schneider, Thomas

Pages

2898

Publication year

2013

Publication date

Dec 2013

Publisher

Nature Publishing Group

e-ISSN

20411723

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1038/ncomms3898

ProQuest document ID

1464667443

Optical sinc-shaped Nyquist pulses of exceptional quality

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