# Frequency-time coherence for all-optical sampling without optical pulse
source

S. Preussler  Insitut für Hochfrequenztechnik, Technische Universität
Braunschweig,Schleinitzstr. 22, 38106 Braunschweig, Germany  G. Raoof Mehrpoor
Insitut für Hochfrequenztechnik, Technische Universität
Braunschweig,Schleinitzstr. 22, 38106 Braunschweig, Germany  T. Schneider
Corresponding author<EMAIL_ADDRESS>
Sampling is the first step to convert an analogue optical signal into a
digital electrical signal. The latter can be further processed and analysed by
well-known electrical signal processing methods. Optical pulse sources like
mode-locked lasers are commonly incorporated for all-optical sampling, but
have several drawbacks. A novel approach for a simple all-optical sampling is
to utilise the frequency-time coherence of each signal. The method is based on
only using two coupled modulators driven with an electrical sine wave,
allowing simple integration in appropriate platforms, such as Silicon
Photonics. The presented method grants all-optical sampling with electrically
tunable bandwidth, repetition rate and time shift.

### Introduction

Analog-to-digital conversion (ADC) is essential for precise transformation and
processing of information carried by electromagnetic signals into their binary
correspondent. In computer technology, the powerful software improvement has
kept pace with more complex hardware development. This is not the case in
electronic systems, since the ADC and sampling rate of signals is a limiting
factor for the performance. By the rapid growth of photonic technologies,
there has been a great interest in reconciling the ADC requirement with
optical components during the last 25 years. The electronic ADC schemes are
limited by several noise sources such as quantisation noise, thermal noise and
nonlinearities 1, 2, motivating further experimental investigations on
photonic counterpart.

According to the well-known Nyquist sampling theorem, to sample and restore a
signal with maximum frequency of $f_{max}$ Hz properly, pulses with a minimum
repetition rate of $2f_{max}$ should be utilised. This complies a sampling
frequency of $f_{s}=2f_{max}=1/t_{s}$, where $t_{s}$ is the time interval
between two sampling points. To achieve high resolution optical sampling,
different methods have been proposed. The simplest method for sampling of an
optical signal is to utilise a fast photodetector in combination with sample-
and-hold circuits. However, the signal is sampled completely in the electrical
domain and therefore restricted by the bandwidth of the electronic components.
Another approach is based on a multiplication between the signal and a train
of short laser pulses (see the Methods for details). The multiplication can be
carried out in a modulator, driven by the electrical signal 1. For this method
the sampling rate is restricted by the bandwidth of the modulator. Faster
sampling is achieved by multiplication in a nonlinear element 3, 4. A temporal
magnifier such as a time lens can be used for very short optical waveforms 5.
All mentioned techniques require a stable short-period optical pulse source.

Conventionally, a mode-locked laser (MLL) is utilised as the pulse source to
perform optical sampling, where the sampling rate depends on the repetition
rate of the MLL. Besides the cost and limited stability of MLLs, they are not
flexible to integrate into an electronic system. The amplitude noise and time
jitter between the pulses are remarkable, affecting the ADC accuracy.
Additionally, the repetition rate of the pulses and therefore the sampling
frequency depends on the optical path length of the MLLs cavity. A tuning of
the sampling frequency is therefore rather complicated and only possible in a
limited range. Furthermore, for a parallelisation of the sampling in time,
tunable delay lines for each branch and an exact calibration is required. For
a parallelisation in the frequency domain, interleaved pulses of different
wavelengths are necessary 6.

In this paper, we demonstrate a simple all-optical sampling, without requiring
any pulse source. All sampling parameters can be simply and fast tuned in the
electrical domain and a parallelisation of sampling in time and frequency
domain is straight forward.

### Theory

The method is based on a convolution between a frequency comb and the signal
spectrum in two coupled modulators. A similar technique can be used for the
demultiplexing of Nyquist channels 7.

Figure 1: Basic principle of optical sampling based on frequency-time
coherence. The insets (a) and (b) show the input signal in the frequency and
time domain, respectively. In the first Mach-Zehnder modulator (MZM1) the
spectrum of the input signal is tripled (inset (c)). The second modulator
leads to the spectral components shown in red in inset (d). In order to adapt
the sampling parameters to the signal, the frequency as well as the phase of
the sinusoidal wave generated by the radio frequency generator (RFG) can be
changed. The result is a convolution of the input spectrum with a nine-line
rectangular frequency comb, corresponding to an optical sampling with a
Nyquist pulse sequence in the time domain (inset e). The sampling point is the
integration of the sequence in the time-interval of the repetition rate, which
can be carried out with a low bandwidth photodiode (PD).

The basic principle of all-optical sampling based on frequency-time coherence
is depicted in Fig. 1. For this sampling just two coupled optical Mach-Zehnder
modulators (MZM) are required. The sinusoidal input frequency $f_{1}$ is
generated by a radio frequency (RF) generator and can be changed in frequency
and phase. This input frequency is tripled in a frequency tripler and used to
drive the first modulator (MZM1). The second modulator (MZM2) is directly
driven with $f_{1}$. The bias voltage and RF power of both modulators are
adjusted such that just two new sidebands with equal power and phase as the
carrier are generated 8. The lengths of the electrical and optical connections
have to be compensated, which can be done by an additional phase shifter
either in the $f_{1}$ or $3f_{1}$ branch (not shown). For the sake of
simplicity, it is assumed that the optical input signal has a band limited
spectrum with a Gaussian-like shape and a frequency and time representation
shown in Fig. 1a and Fig. 1b, respectively. The first modulator simply triples
the spectrum as can be seen from Fig. 1c. The frequency separation between the
spectra is defined by $3f_{1}$. The second modulator leads again to two new
copies of the input spectrum around each carrier. Since three input carriers
are present, the results are the six red copies of the input spectrum, shown
in Fig. 1d. Thus, the output spectrum consists of nine equal copies of the
input spectrum, with a frequency separation of $f_{1}$. Since the modulators
were adjusted in a way that the sidebands have the same amplitude as the
carrier, the nine copies are almost equal to each other. The obtained output
spectrum can be seen as a convolution between a nine-line rectangular
frequency comb and the input spectrum. This convolution corresponds to a
multiplication between a sinc-pulse sequence and the time signal in the time
domain, as can be seen in Fig. 1e. Thus, the result is an optical sampling of
the time domain signal by a sinc-pulse sequence. The same can be expressed
mathematically by the equation:

$\displaystyle\mathcal{F}^{-1}\Big{\\{}X(f)\ast\big{[}\Comb(f)\times\Pi(f)\big{]}\Big{\\}}$
$\displaystyle=x(t)\times(\comb(t)\ast\text{sinc}(t)),$ (1)

where $\mathcal{F}^{-1}$ is the inverse Fourier transform and $\ast$ depicts
the convolution. Here $\comb(t)$ is the time domain representation of an
unlimited frequency comb, $\Comb(f)$,

$\Comb(f)=\sum_{n=-\infty}^{\infty}\delta(f_{c}-nf_{1}),$ (2)

and the rectangular function $\Pi(f)$ is defined along the bandwidth of the
comb in frequency domain

$\Pi(f)=\begin{cases}1&-\frac{Nf_{1}}{2}\leq f_{c}\leq\frac{Nf_{1}}{2}\\\
0&\text{otherwise}.\end{cases}$ (3)

$f_{1}$ denotes the frequency spacing between the comb lines, with $N$ the
total number of lines, determining $B=Nf_{1}$ as the required bandwidth for
the rectangle and $f_{c}$ as the carrier frequency. This definition leads to
an odd number of lines in the comb. For an even number of comb lines, the time
domain representation is a sinc-pulse sequence as well 8.

If the input signal is a Dirac delta function, i.e. $X(f)=\delta(f)$, the time
response would be $x(t)=1$. Therefore, if the input to the two modulators is a
narrow linewidth optical wave, the output would be a rectangular nine-line
frequency comb, or a sinc-pulse sequence in time domain. The repetition time
of the sequence corresponds to the inverse of the frequency spacing between
the lines, or the RF frequency applied to the modulator $t_{s}=1/f_{1}$.
Accordingly, the inverse Fourier transform of a multiplication between a
frequency comb and a rectangular function leads to a sinc-pulse sequence 8, 9,
10:

$\mathcal{F}^{-1}\Big{\\{}\Comb(f)\times\Pi(f)\Big{\\}}=\comb(t)\ast\text{sinc}(t).$
(4)

Due to reciprocity of the Fourier transform, a multiplication in frequency
corresponds to a convolution in time domain. Thus, the right hand side of
equation (4) is the convolution of the two functions in time domain. Having
the function definitions in time domain as

$\displaystyle\comb(t)$
$\displaystyle=\sum_{n=-\infty}^{\infty}\delta(t-n/f_{1}),$ (5)
$\displaystyle\text{sinc}(t)$ $\displaystyle=\dfrac{\sin(\pi t)}{\pi t},$ (6)

the convolution can be written as 8:

$\comb(t)\ast\text{sinc}(t)=\dfrac{\sin(\pi Nf_{1}t)}{N\sin(\pi f_{1}t)}.$ (7)

Generally, the resulting Nyquist sequence of equation (4) is 8

$\dfrac{\sin(\pi Nf_{1}t)}{N\sin(\pi
f_{1}t)}=\sum_{n=-\infty}^{\infty}(-1)^{(N-1)n}\text{sinc}\big{(}\pi
Nf_{1}\big{(}t-\dfrac{n}{f_{1}}\big{)}\big{)}.$ (8)

Thus, the desired Nyquist pulse sequence is the unlimited summation of single
sinc-pulses, each of which is shifted by $n/f_{1}$ and unlimited in time
domain and therefore non-causal. The interference-free summation of these
unlimited pulses is only possible due to the orthogonality of the pulses to
each other, if the time shift corresponds to $k/(Nf_{1})$ where $k$ is an
integer.

Suppose an arbitrary signal $X(f)$ in frequency domain with $x(t)$ being the
signal in the time domain. The convolution of this signal in the frequency
domain with a rectangular frequency comb corresponds to a multiplication
between the signal and the Nyquist pulse sequence in time domain:

$\displaystyle\mathcal{F}^{-1}\Big{\\{}X(f)\ast\big{[}\Comb(f)\times\Pi(f)\big{]}\Big{\\}}$
$\displaystyle=x(t)\times(\comb(t)\ast\text{sinc}(t))$
$\displaystyle=x(t)\times\dfrac{\sin(\pi Nf_{1}t)}{N\sin(\pi f_{1}t)}.$ (9)

The sampled signal $x_{N}(t)$ is achieved by the right-hand side of equation
(Theory). With the order reversed, the Fourier transform of the sampled signal
can be found as

$\displaystyle\mathcal{F}\Big{\\{}x_{N}(t)\Big{\\}}$
$\displaystyle=\mathcal{F}\Big{\\{}x(t)\times(\comb(t)\ast\text{sinc}(t))\Big{\\}}$
$\displaystyle=X(f)\ast\big{[}\Comb(f)\times\Pi(f)\big{]}.$ (10)

Thus, the convolution of the signal spectrum with the rectangular comb through
modulation by the two coupled modulators (as shown in Fig. 1), is equal to a
multiplication of an externally generated sinc-pulse sequence by the time
domain signal (as in Fig. 8).

### Experiment

First, with a CW signal as input, sinc-shaped pulse sequences were generated
as shown in Fig. 2(a). The according frequency domain representation, i.e. the
frequency comb, can be seen in Fig. 2(b). The $9$ comb lines have a frequency
spacing of $10$ GHz. For a single pulse, this corresponds to a bandwidth of
$B=Nf_{1}=90$ GHz (See the dashed rectangle in Fig. 2(b)). The time to the
first zero-crossing is accordingly $\tau=1/B\approx 11$ ps which corresponds
approximately to the full-width at half-maximum duration of the pulses.

(a)

(b)

Figure 2: (a) Generated sinc-shaped pulse sequence in time domain, if a narrow
linewidth optical wave is used as the input to the two modulators; (b)
corresponding frequency domain representation of the $9$-line spectral comb
with $f_{1}=10$ GHz. The dashed rectangle is the bandwidth of one single sinc-
pulse with the same duration. The broad width of the lines is a result of the
limited resolution of the used optical spectrum analyser. The linewidth is a
result of the laser source and in the kHz-range, thus around $6$ orders of
magnitude smaller than the comb bandwidth.

Since both modulators are driven only with two RF frequencies generated from a
single source, the generated sinc-pulse sequence is highly stable and shows a
notably low noise-level. As expected, high quality sinc-shaped pulses are
achieved with time jitter of less than $82$ fs and low amplitude noise
corresponding to a high signal- to-noise ratio level 8. The bandwidth and
repetition time of the pulses can be changed very fast and precisely by an
alteration of the RF input frequency. Additionally, the sampling point can be
shifted in respect to the signal by a phase change of the electrical signal.
Thus, the sampling can be adapted to the signal and it can be scanned through
it electrically. Correspondingly, all-optical sampling of an arbitrary signal
is realised through the introduced approach with fully tunable sampling rate.
Thus, the technique offers flexible sampling of the signals at lower costs,
without requiring pulse sources.

Figure 3: Detailed experimental setup. FL: fiber laser, PC: polarization
controller, MZM: Mach-Zehnder modulator, EDFA: Erbium doped fibre amplifier,
OSO: optical sampling oscilloscope, OSA: optical spectrum analyser.

Figure 3 indicates the basic experimental setup for the proposed all-optical
sampling technique. The optical signal to sample is generated from an
electrical signal in the “Signal Block”. The electrical signal is converted to
the optical domain by a conventional Mach-Zehnder modulator (MZM1). A CW fibre
laser (FL) operating at the wavelength of $1549.72$ nm with $40$ mW output
power generates the optical carrier. Considering power loss caused by the MZM
in the Signal Block the modulated signal is amplified by the first Erbium-
doped fibre amplifier (EDFA1) afterwards. The output power of $17$ dBm is
split in a $3$-dB coupler in order to monitor the desired optical signal at
one channel of an optical sampling oscilloscope (OSO). Subsequently, the
signal is fed to the “Nyquist/Sampling Block” including two coupled MZMs
driven by two sinusoidal RFs generated from one generator. The output signal
of MZM2 is amplified by EDFA2 to $10$ dBm to compensate the power loss during
modulation through MZM3. The RF $f_{1}$ is produced by an RF generator. The
pulsewidth as well as the point where the signal is sampled can be controlled
by this modulation frequency. The first modulator in the Sampling Block (MZM2)
is driven with the tripled frequency $3f_{1}$. The variable phase shifter is
responsible for the time delay of the sinc-pulse train. At a frequency of $10$
GHz it is capable of shifting the phase more than $380$ degrees. Afterwards,
the signal is split. In one branch the frequency is tripled and fed to MZM2.
In the second branch the electrical path difference is adapted by a phase
shifter and fed to MZM3. Additionally, both generators are synchronised with
each other.

Figure 4: Optically sampled 16-bit data packet created by a pattern generator
with data rate of $10$ Gbps. The frequency $f_{1}$, applied to the coupled
modulators, was $10$ GHz.

For the stabilisation of the frequency comb, the bias voltage of the
modulators is controlled in a simple feedback loop. Thus, an optical spectrum
analyser (OSA) measures the maximum amplitudes of the copied spectra and a
computer program adjusts the bias voltage in a way, that all these amplitudes
remain equal. Output powers of the amplifiers are adjusted to constant values
in order to assure the accuracy of the measurement at each step. All
modulators are equipped with polarisation controllers (PC) in order to
compensate the polarisation alterations along the fibres and obtain the
maximum output power. The polarisation is aligned for maximum transmission.
After the Sampling Block, the signal convolved with the frequency comb is
amplified in EDFA3 in order to provide sufficient input power for the OSO.
With the help of a $1$ nm band-pass filter the out-of-band noise components
are reduced.

We have investigated optical sampling of an arbitrary $16$-bit data packet to
prove the performance of the proposed technique. A pattern generator was used
to create the intended data packet with a repetition rate of $100$ ps. The
resulting optically sampled signal can be seen in Fig. 4. For visualisation a
commercial OSO was used. The red curve shows the desired signal, while the
black lines correspond to the sampled signal with a rate of $100$ ps. During
the measurement the OSO was triggered by the sampled waveform. As can be seen,
regardless of the quality of the signal from the pattern generator, it is
multiplied by the pulses and sampled, respectively. Conforming to electronics,
the all-optical sampling can be accomplished for periodic or non-periodic
signals. Since the generated pulse-sequences possess very high quality (see
Fig. 1a), we address the slight differences between the reference and the
sampled signal to trigger problems of the OSO. The point of sampling can be
varied by changing the phase of the RF frequency used to drive the modulators.
Thus, for periodical signals a simple and fast scanning through the signal is
possible by an electrical phase change.

The measurement of a $40$ GHz sinusoidal signal can be seen in Fig. 5, where
the reference trace is represented by the dashed gray line. Therefore, the OSO
was replaced by a photodiode and an oscilloscope. Due to the limited bandwidth
of the photodiode ($50$ GHz) the pulses are distorted. Within each trace of
the measurement two consecutive pulses are shown and analysed. During the
measurement the phase of $f_{1}$ was shifted, leading to a delay of the sinc-
pulse train in steps of $11.1$ ps. Finally, the signal traces were recorded
and integrated by software within a time window of $100$ ps. The result is
illustrated by the red rectangles in Fig. 5 representing the sampling points.

Figure 5: Optical sampling of a $40$ GHz sinusoidal signal. The reference is
displayed by the gray curve, the coloured traces represent the pulse trains at
different phase values and the red dots the integrated, or the sampling
values, respectively. The measurement was carried out with low bandwidth
equipment.

### Discussion

The maximum achievable sampling rate for periodical signals is restricted to
four times the modulator’s bandwidth. Besides, only one modulator driven with
$4$ equidistant input frequencies can be used instead of two modulators (see
the Methods for details). For high-bandwidth non-periodic signals a
parallelisation of the method is required. Then again the sampling rate can be
defined by the bandwidth of the pulses $B=Nf_{1}$ and the sampling frequency
is $f_{s}=B=1/\tau$. This parallelisation can be achieved in time or in
frequency domain. In time domain, the optical input signal is coupled through
a $1:N$ coupler into $N$ parallel branches as can be seen in Fig. 6. In each
branch the signal should be sampled at a fixed time delay compared to the
previous. This time delay can be simply achieved by a phase change between the
electrical sine waves, feeding the Nyquist sources. Thus, no optical tunable
delays are required. Such tunable delay stages would only be possible by a
mechanical tuning or by slow light methods 11.

Figure 6: Principle configuration for parallelisation in time domain. Input
optical signals are fed to a $1:N$ coupler and transferred via optical fibres
to $N$ contiguous Nyquist sequence branches, equally delayed by a phase shift
of the electrical signal. Photodiodes (PD) are utilised for detection of the
sampled optical signal. Of course, the setup can be changed in a way that the
$N$ branches are fed with the Nyquist pulses generated from one source.
Thereby, the multiplication by the signal can be accomplished by a modulator
in each branch and a delay of the optical signal. Red lines are optical and
black lines are electrical connections. The measurement was carried out with
an OSO. Figure 7: Proof-of-concept parallelisation of optical sampling method.
Sinusoidal waveform with frequency of $5$ GHz sampled by convolution in two
coupled modulators. Each coloured trace represents a different sampling
channel, here carried out by an electrical phase shift of $f_{1}$ and
subsequent measurements.

As a substitute, a fast and simple adaptation of all sampling parameters to
the signal is possible, since they are adjusted in the electrical domain. Due
to parallelisation, the method grants $N$ times larger bandwidth with slow
electronic components and photodiodes for non-periodic signals.

In order to demonstrate the parallelisation, in a proof-of-concept experiment
the sampling points were time-shifted by an electrical phase shift of the
frequency $f_{1}$. For every simulated channel a delay of $11$ ps is added
equal to the pulsewidth $\tau$. Different measurements were carried out and
plotted together in Fig. 7.

An alternative method, is to parallelise the sampling signal in frequency
domain through several wavelengths as shown in Fig. 8. The approach is based
on feeding a single Nyquist generator module (two coupled modulators, or one
modulator driven with several coupled radio frequencies) with different input
optical frequencies, causing generation of several frequency combs with
central frequencies equal to the input optical frequencies. The following
dispersion compensating fibre (DCF) delays the sinc-pulse sequences by $2\pi
L\times k_{2}\times\Delta f$, where $L$ and $k_{2}$ are the length and group
velocity dispersion of the DCF, respectively. $\Delta f$ is the frequency
difference between the centre frequencies of the frequency combs. The delayed
sequences are multiplied by the signal in an optical modulator. For higher
sampling speeds, a nonlinear multiplication can be used. The sampling signals
at different wavelengths are demultiplexed in an appropriate device and
detected. In order to reduce costs, the frequency difference could correspond
to a multiple of the ITU-T frequency grid for dense wavelength division
multiplexing ($n\times 50$ GHz) 12. Thus, CW lasers from optical
telecommunications can be used as sources and the demultiplexing can be
performed with an arrayed waveguide grating for instance. The maximum
achievable sampling frequency of the signal, or the sampling rate of the
system, increases with the number of wavelengths, $N$. During parallelisation,
the bandwidth requirements for the electronics remain the same for both
methods. This means, that low bandwidth electronics can sample non-periodic
signals with a bandwidth that is increased by $N$.

In principle this is similar to the time-parallelisation approach, where the
sampling rate is increased with the number of parallel branches. Here the time
delay between the sampling points is controlled by changing the frequency
difference between the optical input waves.

Figure 8: Principle configuration for the parallelisation in frequency domain.
Different input optical inputs with different wavelength ($\lambda_{1}$ to
$\lambda_{N}$ ) are fed to a Nyquist sequence generating block (two coupled
modulators driven with $f_{1}$ and $3f_{1}$, or one modulator driven with
several coupled radio frequencies). The sequences are delayed in a dispersion
compensating fibre (DCF). The multiplication in the time domain is obtained in
an intensity modulator (IM). After demultiplexing of the sampled signals, the
sampling points are detected by parallel photodiodes.

### Conclusion

In conclusion, a method for tunable all-optical sampling without requiring a
pulse source based on time-frequency coherence was presented. Two coupled
modulators were used to convolve the spectrum of the optical input signal with
a rectangular frequency comb, corresponding to a multiplication between the
time domain representation of the signal and a sinc-pulse sequence. The
parameters of the sampling points (sampling rate, bandwidth, position) are
just defined by the sinusoidal input used to drive the modulators. Thus, they
can be fast and easily adapted to the signal and a time shift is possible by a
phase change. The maximum bandwidth of the pulses, or their minimum duration
and therefore the sampling rate is restricted by the bandwidth of the
modulators. With two intensity modulators, the maximum achievable bandwidth is
$4f_{1}$ (see the Methods for details). Thus, with integrated $100$ GHz
modulators 13, a sampling rate of $400$ Gs/s is possible. Sampling rates in
the Tb/s range can be achieved by modulators driven with phase-locked optical
frequencies 10 and by a balanced detector the full field can be sampled 14.

## Supplementary Information

### Sampling

Figure 9: Sampling of a signal in the frequency and time domain with a Dirac-
Delta function (left), corresponding to “ideal” sampling and a sinc-pulse
sequence (right). The $\times$ and $\ast$ denote the signs for a
multiplication and convolution, respectively. The $=$ marks the corresponding
result of the operation, sincs(t) is the sinc-pulse sequence defined in
equation (8)

To convert a continuous to a digital signal in time domain, a sampling of the
time domain waveform is required. The analogue signal to sample $s(t)$ (Fig.
9b) is supposed to be a band limited function with a baseband bandwidth of
$B_{s}$ and the signal can be recovered, if the sampling rate (number of
samples per second) is at least twice the maximum frequency present in the
signal spectrum $f_{s}=1/t_{s}>2B_{s}$. Here $t_{s}$ is the time duration
between two samples, or the inverse of the sampling rate (Fig. 9a, right). The
sampled discrete signal in time domain is (Fig. 9c, right):

$\displaystyle
s_{s}(t)=s(t)\cdot\sum^{\infty}_{n=-\infty}\delta\bigg{(}t-\frac{n}{f_{s}}\bigg{)}$
$\displaystyle=\sum_{n=-\infty}^{\infty}s\bigg{(}\frac{n}{f_{s}}\bigg{)}\delta\bigg{(}t-\frac{n}{f_{s}}\bigg{)},$
(11)

where $\delta(t)$ is an ideal Dirac delta function. In the frequency domain
this corresponds to a convolution ($\ast$) between the frequency domain
representations of the signal $S(f)$ and the Dirac delta function (Fig. 9a,
left):

$\displaystyle S_{s}(f)=S(f)\ast\sum_{n=-\infty}^{\infty}\delta(f-nf_{s})$
$\displaystyle=\sum_{n=-\infty}^{\infty}S(f-nf_{s}).$ (12)

The result is an unlimited number of copies of the signal spectrum with a
frequency distance of $f_{s}$. This sampled signal can be reconstructed by a
multiplication with a rectangular function in the frequency domain (Fig. 9d,
left). This multiplication filters just one of the unlimited number of
spectra.

$S(f)=\sum_{n=-\infty}^{\infty}S(f-nf_{s})\cdot\text{rect}\bigg{(}\frac{f}{f_{s}}\bigg{)}$
(13)

The result is the analogue signal (Fig. 9e, right):

$\displaystyle s(t)$
$\displaystyle=\Bigg{[}\sum_{n=-\infty}^{\infty}s\bigg{(}\frac{n}{f_{s}}\bigg{)}\delta\bigg{(}t-\frac{n}{f_{s}}\bigg{)}\Bigg{]}\ast\text{sinc}(f_{s}t)$
(14)
$\displaystyle=\sum_{n=-\infty}^{\infty}s\bigg{(}\frac{n}{f_{s}}\bigg{)}\text{sinc}\Bigg{(}f_{s}\bigg{(}t-\frac{n}{f_{s}}\bigg{)}\Bigg{)},$

with sinc as the sinus cardinals or sinc-function as defined above. Thus, a
band limited analogue signal can be seen as a superposition of an unlimited
number of sinc-pulses weighted with the sampling value and time shifted by the
sampling time.

The sampling with the method presented here leads in the frequency domain to
(Fig. 9h, left):

$S_{s}(f)=S(f)\ast\Bigg{[}\sum_{n=-\infty}^{\infty}\delta(f_{c}-nf_{s})\cdot\text{rect}\bigg{(}\frac{f-f_{c}}{B}\bigg{)}\Bigg{]}$
(15)

with $f_{c}$ as the carrier frequency of the optical wave. Thus, the copies of
the spectrum are not unlimited as for the sampling with a Dirac Delta
function, but restricted to the bandwidth $B=Nf_{1}=Nf_{s}$. For a
parallelised sampling it can be written that $B=f_{1}=f_{s}$. Therefore, the
sampled signal corresponds to just one copy of the spectrum. Thus, in time
domain it follows (Fig. 9h, right):

$\displaystyle s_{s}(t)=$ $\displaystyle
s(t)\cdot\Big{[}\sum_{n=-\infty}^{\infty}\delta(t-\frac{n}{f_{s}})\ast\text{sinc}(Bt)\Big{]}$
$\displaystyle=$ $\displaystyle
s(t)\cdot\sum_{n=-\infty}^{\infty}\text{sinc}\big{(}B(t-\frac{n}{f_{s}})\big{)}$
$\displaystyle\approx$
$\displaystyle\sum_{n=-\infty}^{\infty}s\big{(}\frac{n}{f_{s}}\big{)}\text{sinc}\big{(}B(t-\frac{n}{f_{s}})\big{)}.$
(16)

By the comparison with equation (14) it can be seen that the method grants
access to the sampling values $s(n/f_{s})$ by an integration of the sinc-pulse
sequence over the time interval $1/f_{1}$. This integration can be carried out
with a photodiode with a restricted bandwidth of $f_{1}=B/N$.

### Tunability

According to the left hand side of equation (4), the frequency representation
of the sinc-pulse sequence is a rectangular frequency comb. The individual
comb lines have to have the same or a linear dependent phase. A simple method
to generate such a frequency comb is to use two coupled intensity modulators.
Both modulators are adjusted in a way that the first order sidebands have the
same amplitude as the carrier and higher order sidebands are suppressed. If
the DC bias and the RF signal voltages applied to a single modulator are
$V_{B}$ and $v_{s}\cos(\omega_{s}t)$, respectively, the expression for the
output field can be written as 15

(a)

(b)

(c)

Figure 10: (a) 9-line frequency comb, the $3$ blue spectral lines are
generated by the first modulator, whereas the second modulator generates the
red lines; (b) Frequency comb with $6$ lines. The first modulator (blue lines)
operates in the suppressed carrier regime (dashed blue line) and the second
modulator generates the red lines; (c) Both modulators operate in the
suppressed carrier regime (dashed lines). The result is again a $6$ line
frequency comb.

$\begin{split}E(t)=\sum_{n=-\infty}^{\infty}(-1)^{k}\big{\\{}\cos(\pi\epsilon)J_{2k}(\pi\alpha)\cos(\omega_{0}t+2k\omega_{s}t)\\\
+\sin(\pi\epsilon)J_{2k-1}(\pi\alpha)\cos(\omega_{0}t+(2k-1)\omega_{s}t)\big{\\}},\end{split}$
(17)

where $J_{k}$ is the Bessel function of the first kind and order $k$,
$\epsilon=V_{B}/V_{\pi}$ and $\alpha=v_{s}/V_{\pi}$, in which $V_{\pi}$ is the
half-wave voltage of the modulator. Contrary to phase modulators, the
amplitude of the carrier, first-order sidebands and higher-order sidebands in
intensity modulators can be adjusted by two parameters, i.e. the RF driving
voltage $\alpha$ and the DC bias $\epsilon$. If equation (17) is reduced to
the spectral components of interest, i.e. the carrier and the first order
sidebands, the electric field at the output of the modulator can be written as
8:

$\begin{split}E(t)=\cos(\pi\epsilon)J_{0}(\pi\alpha)\cos(\omega_{0}t)-\sin(\pi\epsilon)J_{1}(\pi\alpha)\\\
\big{\\{}\cos((\omega_{0}-\omega_{s})t)+\cos((\omega_{0}+\omega_{s})t)\big{\\}}.\end{split}$
(18)

If two intensity modulators are used and both have just one input frequency,
then there are three possibilities of generating a rectangular frequency comb
as illustrated in Fig. 10.

If the carrier is not suppressed as in Fig. 10(a), the modulators generate $9$
lines. The first modulator is driven with the frequency $f_{a}$. Thus, the
modulator generates the three blue lines. In the second modulator these lines
are re-modulated with the frequency $f_{b}=\frac{1}{3}f_{a}$ to produce the
red lines. If at least one of the modulators can be driven with a maximum
frequency of $f_{a}=40$ GHz, the maximum bandwidth is $B=N\times f_{1}=N\times
f_{b}=120$ GHz, which corresponds to a pulse width (FWHM) of around $8$ ps. In
Fig. 10(b) the carrier of the first modulator is suppressed. Thus, an even
number of lines ($6$) is produced. Here the second modulator has to be driven
with $f_{b}=\frac{2}{3}f_{a}$ and the maximum bandwidth is $N\times f_{b}=160$
GHz, corresponding to $6$ ps pulses. If both modulators are driven in the
suppressed carrier regime, again an even number of lines is produced but now
with a maximum bandwidth of $B=60$ GHz ($17$ ps) since
$f_{b}=\frac{1}{4}f_{a}$. Of course the first modulator can be driven with the
lower and the second with the higher frequency, however the results are the
same.

The repetition period of the Nyquist pulses, and therefore the number of
parallel branches or the wavelength for sampling, is defined by the number of
lines. Nine lines as in Fig. 10(a) can be produced if just one modulator is
driven with a four-frequency input signal. Thus, each odd number $N$ of lines
can be produced with just one modulator if the RF-signal driving the modulator
consists of $(N-1)/2$ equally spaced frequency lines, for an even number of
lines the number of RF is $N/2$. These equally spaced frequency lines in the
RF domain can be produced by a mixing of higher harmonics, or by an arbitrary
waveform generator for instance. However, the maximum bandwidth of this method
is restricted by around twice the bandwidth of the used modulator. Another
possibility to increase the number of lines is the usage of more than two
modulators. However, the maximum bandwidth which can be achieved does not
increase much with the number of additional modulators. Higher bandwidths can
be achieved if higher order sidebands are used. Although, since the carrier
and the first order sidebands are out of phase, this requires that these
components are effectively suppressed. One possibility to achieve Nyquist
pulses with very broad bandwidths is the filtering of the frequency comb of a
mode-locked source 11. However, due to the non-ideal bandwidth of optical
filters, the generated pulses have no sinc shape and the pulse parameters
cannot be adapted easily to the sampled signal. Instead of filtering a whole
comb as rectangular as possible, single comb lines can be filtered. These
lines are the initial lines for a subsequent modulation. The initial lines are
responsible for the arbitrary bandwidth, not limited by modulators, and the
subsequent modulation enables an arbitrary repetition period of the pulses. As
a mode-locked source, a fibre fs-laser or much cheaper and simpler, a comb
generator can be used. In proof-of-concept experiments, sinc-shaped Nyquist
pulses with a bandwidth of around $300$ GHz have been generated by this method
10. However, bandwidths in the THz-range and correspondingly femtosecond
pulses are possible from a comb generator.

## Acknowledgements

We acknowledge the funding from the Deutsche Forschungs Gemeinschaft (DFG)
(SCHN716/13-1).

## Author contributions

S.P. and G.R.M. performed the experiments. T.S. proposed the first concept of
the optical source free sampling and the theory. All authors contributed
equally in writing the manuscript.

## Additional information

Competing financial interests: The authors declare that they have no competing
financial interests.

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