Se ha descubierto un modo de multiplicar por diez el rendimiento de las
fibras ópticas. La solución, simple pero imaginativa, reduce la cantidad
de espacio necesario entre los pulsos de luz que transportan los datos,
lo que permite un aumento drástico en la capacidad de las fibras
ópticas.
Las fibras ópticas transportan los datos en forma de
pulsos de luz a través de distancias de miles de kilómetros a
velocidades tremendas. Son una de las glorias de la tecnología moderna
de telecomunicaciones. Sin embargo, tienen capacidad limitada, debido a
que en la fibra hay que alinear los pulsos de luz uno tras otro,
separados por una distancia que no puede ser inferior a cierto límite
mínimo, a fin de garantizar que las señales no se interfieran entre sí.
Esto hace que en la fibra existan espacios vacíos que no se aprovechan
para enviar datos.
Desde su aparición en la década de 1970, la
capacidad de trasmisión de datos de la fibra óptica se ha incrementado
cada cuatro años en un factor de diez, un hecho impulsado por el flujo
constante de nuevas tecnologías que se ha mantenido durante bastante
tiempo. Sin embargo, en los últimos años se ha llegado a un cuello de
botella, y científicos de todo el mundo han estado tratando de salir de
él.
El equipo
de Luc Thévenaz y Camille Brès, de la Escuela Politécnica Federal de
Lausana (EPFL) en Suiza, ha ideado un método para agrupar más los pulsos
en las fibras, sin que ello origine problemas, reduciendo así el
espacio entre los pulsos. Su enfoque hace posible el uso de toda la
capacidad de una fibra óptica. Esto permitirá aumentar diez veces el
rendimiento de los sistemas de telecomunicaciones basados en fibra
óptica.
Optical sinc-shaped Nyquist pulses of exceptional quality
ABSTRACT
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.
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 year1, new approaches are being developed2.
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−1Hz−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 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 efficiency 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 of
26 Tbit s−1 and a net spectral efficiency 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 efficiency 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 fibre 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:
where τp is the pulse duration between zero crossings and β is known as a roll-off factor17, which is in the range 0≤β≤1. 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 transmission12, 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 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 elsewhere12, 13, 14, 15, 16, 17, 18, here the focus is placed on the generation of a sinc-pulse shape as ideal as possible.
Figure 1: Possible multiplexing of sinc-shaped Nyquist pulses.
Periodic sinc-pulse sequences can be split into
branches, each of which corresponds to an independent channel. In the nth branch, the periodic sequence is delayed by n times the interval τ=1/(NΔf), 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.
The temporal and spectral features of sinc-shaped pulses bring
benefits not only to optical communications but also to many other
fields. 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 filters25, 26, 27 with tunable passband profiles, 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 offline to create
Nyquist filtering of the baseband signal. This can provide a quite good
roll-off 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 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 fibre
optical parametric amplification, 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 β=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 (β~0)
is proposed and demonstrated. The method is based on the direct
synthesization of a flat phase-locked frequency comb with high
suppression of out-of-band components. It is theoretically demonstrated
and experimentally confirmed that this comb corresponds to a periodic
sequence of time-unlimited sinc pulses. The wide tunability of the
method, using a proof-of-concept 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 roll-off, 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
high-capacity 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 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 time–frequency 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 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 single-pulse spectrum (see lower figures 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 satisfies the
zero-ISI Nyquist criterion, similar to the ideal single sinc-shaped
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 zero-crossing pulse duration τp=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 requirements35, 36, or to change the parameters of optical sampling devices24 whenever required.
Figure 2: Time–frequency 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 figure shows
the intensity of the time-domain traces instead of the field 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).
Theory
The Nyquist criterion for a pulse y(t) satisfying zero ISI implies that, for a particular sampling period τ=τp/2, y(nτ) is 0 for any non-zero integer n, while y(0)≠0. This means that when the signal is periodically sampled with a period τ, a non-zero value is obtained only at the time origin11. 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 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 flat 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 field of a frequency comb with N lines, having the same amplitude E0/N and frequency spacing Δf around the central frequency f0, 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 sinc-shaped 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 sinc-pulse NΔfsinc(NΔft).
The temporal dependence of the above expression can then be obtained by
taking its inverse Fourier transform and using the Poisson summation
formula37:
where ⊗ denotes the convolution operation. Thus, it follows for the right-hand 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(ts)=1 at every sampling instant
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:
Figure 3: Normalized field 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 π-phase-modulated pulses.
Consequently, it is proven that the field envelope of the time-domain representation of a frequency comb of N identical and equally spaced lines corresponds to an infinite summation of sinc-shaped Nyquist pulses with period
and zero-crossing pulse width
. Thus, considering that the pulse repetition period
is a multiple of the time interval
, the resulting time-domain 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 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-fibre38, 39, Yb-fibre40 and Ti:sapphire41 mode-locked lasers, or from a continuous wave optical source exploiting Kerr-nonlinearities in an optical resonator42, 43, 44, 45, or employing a combination of strong intensity and phase modulation46, 47, 48 together with chirped Bragg gratings49, dispersive medium50 or highly nonlinear fibres51, 52, 53.
However, every comb does not necessarily result in a sequence of
Nyquist pulses, since a sinc-pulse 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 out-of-band lines. Thus, although flat
frequency combs can be obtained using different methods, as for
instance through phase modulation48, 49, 50,
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 comb38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53 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 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 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 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 time-domain pulse train waveforms. While the first
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 satisfied without any carrier suppression, resulting in a frequency spacing between the lines of Δf=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 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 2f1=3f2, giving a line spacing Δf=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 Δf=2f2.
Figure 4: Basic experimental implementation.
(a) Proof-of-concept set-up.
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=Δf. (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=Δf/2.
A high-quality rectangular-shaped frequency comb can be obtained by tuning the DC bias VB and the RF voltage amplitude νs
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 νs 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 sinc-pulse sequence into the Nyquist bandwidth, a low modulating voltage νs must be used to strongly suppress the out-of-band components. In particular, the RF-driving voltage νs of both modulators is here adjusted to remain below ~0.36Vπ (where Vπ
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
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 sinc-shaped Nyquist pulses
The
quality of the pulses and the flexibility of the method have been
experimentally verified 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 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 (red-dashed 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 f1=30 MHz and f2=Δf=10 MHz, resulting in sinc-shaped Nyquist pulses with zero-crossing pulse duration of τp=22.22 ns, full-width at half-maximum (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.
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=Δf=10 MHz, (b) f1=300 MHz and f2=Δf=100 MHz, (c) f1=3 GHz and f2=Δf=1 GHz, and (d) f1=30 GHz and f2=Δf=10 GHz. The maximum difference between measured pulses and theoretical ones remained in all cases below 1%.
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 flat 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.
Figure 6: Frequency and time-domain representation of the generated sinc-shaped Nyquist pulses.
Pulse duration and repetition rate can
be easily modified 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 Δf=10 GHz, and expanding over a bandwidth of 90 GHz. (c) Spectrum and (d) time-domain waveform of a comb generated with N=10, Δf=10 GHz, and bandwidth of 100 GHz. (e) Spectrum and (f) time-domain waveform of a comb generated with N=15, Δf=6 GHz, and bandwidth of 90 GHz. (g) Spectrum and (h) time-domain waveform of a comb with N=6, Δf=26
GHz, and an extended bandwidth of 156 GHz. The comb has been spectrally
broadened using the second-order sidebands of the first 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.
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 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 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 zero-crossing pulse duration has been reduced down to τp=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 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 higher-order sidebands, as reported in Fig. 6e. The measured sinc pulse has a zero-crossing duration of τp=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 νs≈1.52 Vπ
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 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 Δf=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 τp=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 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 (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 roll-off 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 roll-off 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
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 β=0 is compared with the roll-off obtained by other optical pulse-shaping methods13, 18, 20, 21 (reporting β=0.4 in the best case21),
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 time-domain measurements shown in Figs 5 and 6 with results reported in refs 13, 18, 20 and 21.
Figure 7: High stability and quality of periodic sinc pulses.
(a) Colour-grade figure for one of the measured sinc-pulse sequences. In this case, a frequency comb with N=9 spectral components separated by Δf=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
SNR>40 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 β. The r.m.s. error is minimized for β=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.
In 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
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
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 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 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/Δf, the frequency comb will be convolved with a sinc function in the frequency domain9 having zero crossings at n·1/T=n·Δf, 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 verified by modulating the generated sequence of
sinc-shaped pulses using a pseudo-random binary sequence with a length
of 231–1. 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 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
non-ideal rectangular modulation window and additional convolutions
between the very small out-of-band comb lines and the modulation
spectrum.
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 finite 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 filtering 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.
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, confining
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 (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 proof-of-concept set-up, is
significantly lower than the one obtained by other optical pulse-shaping
methods13, 18, 21. Such methods actually report a roll-off 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 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.
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 benefits
for optical spectroscopy, all-optical sampling devices and photonic
analogue-to-digital converters, among other potential applications.
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 (carrier and two first-order sidebands) and the remaining M−m modulators each produces two lines (two first-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 field from each modulator has to be analyzed. If the DC bias and
the RF signal voltages applied to a single modulator are VB and νs cos(ωst), respectively, its normalized output optical field is given by the expression48, 56:
where Jk is the Bessel function of the first kind and order k, =VB/Vπ, and α=vs/Vπ, in which Vπ
is the half-wave voltage of the modulator. Note that according to
equation (12), the amplitude of the carrier, first-order sidebands and
higher-order sidebands can be adjusted by a proper tuning of the
RF-driving voltage α and the DC bias ∈. The primary
objective is to equalize the amplitudes of the carrier and first-order
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 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 significant
difference with respect to the use of phase modulators46, 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
uniform-locked phase. Figure 9a shows a contour plot representing the amplitude difference between the first-order sidebands and the carrier (that is, −J1(πα/2)sin(π∈/2)−J0(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 first-order
sidebands. Actually, as depicted in Fig. 9a, the relation between the optimum bias voltage VB and the driving RF signal amplitude vs that fulfils this condition is a periodic function, which can be simply obtained from equation (13):
Figure 9: Rectangular frequency comb generation with MZMs.
(a) Amplitude difference between first-order sidebands and carrier component [−J1(πα/2)sin (π∈/2)−J0(πα/2)cos (π∈/2)]. Equalization of the amplitude between the two first-order sidebands and carrier is only possible if pairs of bias voltage ∈ and driving voltage α
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
first-order sideband amplitudes. Power levels have been normalized to
the maximum power reached by the first-order sidebands.
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
first-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 α,
when the optimum bias is set according to equation (14). It can be
observed that a high amplitude of the first-order sidebands (equal to
the carrier amplitude) together with a low amplitude of higher-order
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 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
confine the sinc-pulse sequence into the Nyquist bandwidth. 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 figure
have been normalized to the maximum power of the equalized first-order
sidebands). The figure 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 justified as a result from the asymptotic form of the Bessel function Jk(x)~xk for small argument x.
According to Fig. 9c, the maximum power of the carrier and the first-order sidebands can be reached using a driving voltage vs=0.8Vπ.
This condition offers a 15-dB suppression of the second-order sidebands
(see red-dashed line in the figure). 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 first-order sidebands. Thus, for instance, using a modulating
voltage vs=0.32Vπ, 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 first-order
sidebands with respect to the maximum reachable power level. Thus,
arbitrary out-of-band suppression can be obtained using lower RF
voltages, while the power reduction of carrier and first-order sidebands
can be easily compensated by optical amplification.
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=∈Vπ, ∈
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 in equation (12)
vanishes. This optimal condition is given by a driving voltage vs≈1.52Vπ. 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 Vπ 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.
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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branches, each of which corresponds to an independent channel. In the nth branch, the periodic sequence is delayed by n times the interval τ=1/(NΔf), 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.
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.
, 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 follows:
that is 1 for all integers n where
and 0 elsewhere, the above equation can be written as:
, covering a bandwidth NΔf, is represented in the time domain by the sinc-pulse NΔfsinc(NΔft).
The temporal dependence of the above expression can then be obtained by
taking its inverse Fourier transform and using the Poisson summation
formula37:
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(ts)=1 at every sampling instant
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:
and zero-crossing pulse width
. Thus, considering that the pulse repetition period
is a multiple of the time interval
, the resulting time-domain envelope x(t) satisfies the following condition for any integer m:
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