Photonic arbitrary waveform generation based on the temporal Talbot effect

Hao Chi; Shanyi Wang; Shuna Yang; Yanrong Zhai; Xihua Zou; Bo Yang; Qiliang Li

doi:10.1364/OE.425209

1. Introduction

The synthesis of arbitrary waveforms with a bandwidth above gigahertz is vital to many modern applications in different fields, such as wireless and optical communications, radar, medical imaging, electronic countermeasure and quantum computing [1–3]. However, the electronic approach for arbitrary waveform generation (AWG) is limited by the sampling rate of digital-to-analog converters. In recent decades, photonic approaches for AWG have attracted much research interest as the photonic techniques feature the inherent advantage of wide bandwidth, and many photonic AWG methods have been proposed [4–15]. The approaches of Fourier-domain pulse shaping and direct space-to-time pulse shaping are based on the modulation of spatially-dispersed optical spectrum [4–8], where a bulk spectral shaper based on a spatial light modulator (SLM) should be employed. The Fourier-domain pulse shaping can also be implemented in the time domain [9–11], where a pair of dispersion elements with conjugate dispersion values are necessary. Photonic AWG based on fiber-optic spectral shaping and wavelength-to-time mapping has been demonstrated in [12–15]. However, it is a challenging task to realize reconfigurable frequency response by using a fiber-optic platform, which usually involves complicated transversal multi-tap fiber-delay-line networks.

In this paper, we present a novel photonic approach for AWG based on the real-time Fourier transform in the temporal Talbot effect [16–18]. In the approach, a periodic pulse train is firstly modulated by a multi-tone sinusoidal signal and then passes through a dispersive element with a dispersion value that satisfies the temporal Talbot effect. The output pulse train has multiplied pulses and an envelope decided by the spectrum of the modulating multi-tone signal. It is shown that common double-sideband (DSB) modulation can be employed to achieve the generation of symmetric waveforms and single-sideband (SSB) modulation can be used to generate asymmetric waveforms. The major advantage of the approach lies in its simple structure, where only a time-domain modulator is required and a multi-tone signal is sufficient for AWG. We can obtain any desired waveforms by injecting a predetermined multi-tone RF signal with appropriate frequencies and powers. In addition, there is no requirement on the synchronization between the pulses and the injected RF signal. A concise and strict theoretical framework with a closed-form expression on the final time-domain waveform is presented, for the first time to the best of our knowledge, which exactly explains the relationship of real-time Fourier transform between the pulse trains before and after the dispersion element. A proof-of-concept experiment has been implemented to verify the concept of the approach and the correctness of the theoretical findings. Some numerical results are presented to further demonstrate the proposed AWG approach.

2. Principle

A schematic diagram of the proposed AWG method is shown in Fig. 1. In the system, an input pulse train is firstly amplitude-modulated by an analog signal $x(t)$ and then passes through a dispersive medium with a proper dispersion value determined by the temporal Talbot effect. The output pulse train after the dispersion medium in the time domain is a time mapped version of the spectrum of the signal $x(t)$, which has been employed to achieve real-time spectral analysis [18–19]. In this work, we propose to employ a multi-tone RF signal, composed of a number of sinusoidal signals with specific frequencies and amplitudes, to generate a desired waveform. Firstly, we will discuss the relationship between the modulating signal and the output waveform using a concise theoretical model based on the principle of the temporal Talbot effect.

Fig. 1. Arbitrary waveform generation based on the real-time Fourier transform in the temporal Talbot effect. EOM: Electro-optic modulator.

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The electric field of an input pulse train with a period of $T$ is expressed as ${g_T}(t) = \sum\limits_{n ={-} \infty }^\infty g (t - nT)$, where $g(t)$ denotes the individual pulse in a period. We assume the pulse train has a low duty cycle; i.e., the time duration of $g(t)$ is much less than the period $T$. An analog signal $x(t)$ is then modulated on the pulse train via an electro-optic modulator (EOM). Note that the bandwidth of $x(t)$ and the repetition rate of the pulse train ${g_T}(t)$ satisfy the Nyquist criterion. The modulated pulse train ${e_M}(t) = x(t){g_T}(t)$ equals $x[n]{g_T}(t)$, where $x[n]\textrm{ = }x(nT)$, as $x(t)$ is slowly varying compared with $g(t)$. Therefore, ${e_M}(t)$ can be further expressed as [20–21]

(1)$${e_M}(t) = x[n]{g_T}(t) = \sum\limits_{n ={-} \infty }^\infty {x[n]g} (t - nT) = x[n] \cdot \{ g(t) \ast {\delta _T}(t)\} = g(t) \ast {x_T}(t)$$

where ${\ast} $ denotes the convolution operation, ${\delta _T}(t) = \sum\limits_{n ={-} \infty }^\infty \delta (t - nT)$ is the periodic impulse train (Dirac comb), ${x_T}(t) = x(t) \cdot {\delta _T}(t) = \sum\limits_{n ={-} \infty }^\infty {x(t)\delta } (t - nT)$ is the sampled signal.

Let’s first consider the sampled signal ${x_T}(t)$ propagating through a dispersion element with a dispersion value $\ddot{\Phi }$ that satisfies the condition $|\ddot{\Phi }|= {T^2}/2\pi $ [20]. The impulse response of the dispersion medium is as $h(t) = \exp (j\frac{{{t^2}}}{{2\ddot{\Phi }}})$. The optical signal after the dispersion element can be expressed as

(2)$$\begin{aligned} s(t) &= {x_T}(t) \ast h(t) = {x_T}(t) \ast \exp (\frac{j}{{2\ddot{\Phi }}}{t^2}) = \int_{ - \infty }^{ + \infty } {d\tau \cdot {x_T}(\tau )} \exp [\frac{j}{{2\ddot{\Phi }}}{(t - \tau )^2}]\\ &= \exp (\frac{j}{{2\ddot{\Phi }}}{t^2})\int_{ - \infty }^{ + \infty } {d\tau \cdot {x_T}(\tau )} \exp (\frac{j}{{2\ddot{\Phi }}}{\tau ^2})\exp ( - j\frac{t}{{\ddot{\Phi }}}\tau ) \end{aligned}$$

The integral in the above equation can be treated as the Fourier transform, where the term $\exp( - j\frac{t}{{\ddot{\Phi }}}\tau )$ is regarded as the kernel of the Fourier transform, the variables of which are $\tau$ and $\omega = t/\ddot{\Phi }$. Therefore, Eq. (2) is re-written as

(3)$$\begin{aligned} s(t) &= \exp (\frac{j}{{2\ddot{\Phi }}}{t^2}) \cdot {\left. {{\mathfrak{F}}\{ {x_T}(\tau )\exp (\frac{j}{{2\ddot{\Phi }}}{\tau^2})\} } \right|_{\omega = t/\ddot{\Phi }}}\\ &= \frac{1}{{2\pi }}\exp (\frac{j}{{2\ddot{\Phi }}}{t^2}) \cdot \{ { {{\mathfrak{F}}\{ {x_T}(\tau )\} } |_{\omega = t/\ddot{\Phi }}} \ast {\left. {{\mathfrak{F}}\{ \exp (\frac{j}{{2\ddot{\Phi }}}{\tau^2})\} } \right|_{\omega = t/\ddot{\Phi }}}\} \end{aligned}$$

where ${\mathfrak{F}}\{{\cdot} \} $ denotes the operation of Fourier transform. The term ${ {{\mathfrak{F}}\{ {x_T}(\tau )\} } |_{\omega = t/\ddot{\Phi }}}$ can be expressed as

(4)$${ {{\mathfrak{F}}\{ {x_T}(\tau )\} } |_{\omega = t/\ddot{\Phi }}} = {\left. {{\mathfrak{F}}\{ x(t)\sum\limits_{n ={-} \infty }^\infty \delta (t - nT)\} } \right|_{\omega = t/\ddot{\Phi }}} = \frac{1}{T}\sum\limits_{k ={-} \infty }^\infty X (\frac{t}{{\ddot{\Phi }}} - k\frac{{2\pi }}{T})$$

where $X(\omega )$ is the Fourier transform of $x(t)$. Notice that the term $\sum\limits_{k ={-} \infty }^\infty X (\frac{t}{{\ddot{\Phi }}} - k\frac{{2\pi }}{T})$ denotes a periodic waveform with a period of $2\pi |\ddot{\Phi }|/T = T$. The term ${\left. {{\mathfrak{F}}\{ \exp (\frac{j}{{2\ddot{\Phi }}}{\tau^2})\} } \right|_{\omega = t/\ddot{\Phi }}}$ can be expressed as

(5)$${\left. {{\mathfrak{F}}\{ \exp (\frac{j}{{2\ddot{\Phi }}}{\tau^2})\} } \right|_{\omega = t/\ddot{\Phi }}} = \sqrt {j2\pi \ddot{\Phi }} {\left. { \cdot \exp ( - j\frac{{\ddot{\Phi }}}{2}{\omega^2})} \right|_{\omega = t/\ddot{\Phi }}} = T\exp ( - \frac{j}{{2\ddot{\Phi }}}{t^2})$$

where we have applied the condition $\textrm{|}\ddot{\Phi }\textrm{|= }{T^2}/2\pi $ and neglected the common phase factor $\sqrt j $ as it has no impact on the envelope of the output signal. It is crucial that the convolution between $\sum\limits_{k ={-} \infty }^\infty X (\frac{t}{{\ddot{\Phi }}} - k\frac{{2\pi }}{T})$ and $\exp ( - \frac{j}{{2\ddot{\Phi }}}{t^2})$ can be regarded as that the waveform experiences a dispersion element with a dispersion value $- \ddot{\Phi }$. It is observed that the period of the waveform and the dispersion value satisfy the condition of the integer-order temporal Talbot effect, which means the output waveform is the same as the input one except that there is a T/2 relative delay according to the principle of the temporal Talbot effect [20]. Therefore, Eq. (3) can be simplified as

(6)$$s(t) = \frac{1}{{2\pi }}\exp (\frac{j}{{2\ddot{\Phi }}}{t^2}) \cdot \sum\limits_{k ={-} \infty }^\infty X (\frac{t}{{\ddot{\Phi }}} - k\frac{{2\pi }}{T} - \frac{T}{{2\ddot{\Phi }}})$$

It is seen from Eq. (6) that the output waveform is a scaled, shifted, superposed and time-mapped version of $X\textrm{(}\omega )$ (the Fourier transform of $x(t)$) in each period T, within a quadratic phase factor. If the analog signal is a monotone signal $\cos ({\omega _1}t)$, where ${\omega _\textrm{1}}$ is the angular frequency, the modulating signal $x(t)$, with taking account into the dc bias in the modulation, is as $1 + {\alpha _1}\cos ({\omega _1}t)$, where ${\alpha _\textrm{1}}$ denotes the modulation coefficient. Note that Eq. (6) is equivalent to the Eq. (13) in [18]. The spectrum of $x(t)$ is $2\pi \delta (\omega ) + \pi {\alpha _1}[\delta (\omega - {\omega _1}) + \delta (\omega + {\omega _1})]$. Therefore, the output signal can be obtained as

(7)$$\begin{aligned} s(t) &= \sum\limits_{k ={-} \infty }^\infty {\{ \exp (j{\varphi _0}) \cdot \delta (t - kT - \frac{1}{2}T) + \frac{{{\alpha _1}}}{2}[\exp (j{\varphi _{ + {\omega _1}}}) \cdot \delta (t - kT - \frac{1}{2}T - \ddot{\Phi }{\omega _1})} \\ &\quad + \exp (j{\varphi _{ - {\omega _1}}}) \cdot \delta (t - kT - \frac{1}{2}T + \ddot{\Phi }{\omega _1})]\} \end{aligned}$$

where ${\varphi _0} = {(k + 1/2)^2}\pi $, ${\varphi _{ {\pm} {\omega _1}}} = {\varphi _0} + \frac{1}{{4\pi }}\omega _1^2{T^2} \pm (k + \frac{1}{2}){\omega _1}T$ are phase terms. Since the optical signal before the dispersion element is $g(t) \ast {x_T}(t)$ as in Eq. (1), the real output signal should be the convolution of $g(t)$ and $s(t)$, that is ${e_{out}}(t) = g(t) \ast s(t)$, in which the pulse $g(t)$ replaces the delta functions in Eq. (7), as

(8)$$\begin{aligned} {e_{out}}(t) &= \sum\limits_{k ={-} \infty }^\infty {\{ \exp (j{\varphi _0}) \cdot g(t - kT - \frac{1}{2}T) + \frac{{{\alpha _1}}}{2}[\exp (j{\varphi _{ + {\omega _1}}}) \cdot g(t - kT - \frac{1}{2}T - \ddot{\Phi }{\omega _1})} \\ &\quad + \exp (j{\varphi _{ - {\omega _1}}}) \cdot g(t - kT - \frac{1}{2}T + \ddot{\Phi }{\omega _1})]\} \end{aligned}$$

We can see from Eq. (8) that there are three pulses corresponding to each original pulse, which relates to the carrier, upper-and lower-sidebands. The carrier in the modulated signal (dc component), if there is any, corresponds to the pulse at $t = kT + T/2$ in each period. Moreover, the intensity of the sideband pulses is proportional to the modulation coefficient ${\alpha _1}$ and the time interval between a carrier pulse and a sideband pulse is proportional to the frequency ${\omega _\textrm{1}}$. Therefore, we can use a RF signal with multiple frequencies and different powers as $x(t) = \sum {{\alpha _1}\cos ({\omega _1}t)} $ to generate a desired waveform. Because it’s a linear process, in the same way, the result of modulating the two frequencies can be expressed as

(9)$$\begin{aligned} {e_{out}}(t) &= \sum\limits_{k ={-} \infty }^\infty {\{ \exp (j{\varphi _0}) \cdot g(t - kT - \frac{1}{2}T)} \\ &+ \frac{{{\alpha _1}}}{2}[\exp (j{\varphi _{ + {\omega _1}}}) \cdot g(t - kT - \frac{1}{2}T - \ddot{\Phi }{\omega _1}) + \exp (j{\varphi _{ - {\omega _1}}}) \cdot g(t - kT - \frac{1}{2}T + \ddot{\Phi }{\omega _1})]\\ &+ \frac{{{\alpha _2}}}{2}[\exp (j{\varphi _{ + {\omega _2}}}) \cdot g(t - kT - \frac{1}{2}T - \ddot{\Phi }{\omega _2}) + \exp (j{\varphi _{ - {\omega _2}}}) \cdot g(t - kT - \frac{1}{2}T + \ddot{\Phi }{\omega _2})]\} \end{aligned}$$

where ${\alpha _2}$ represent for the modulation coefficient, ${\omega _2}$ represent for the frequency of the other RF signal and ${\varphi _{ {\pm} {\omega _2}}} = {\varphi _0} + \frac{1}{{4\pi }}\omega _2^2{T^2} \pm (k + \frac{1}{2}){\omega _2}T$.

Due to the symmetry in the spectrum of a modulated signal with double sideband modulation (DSB), common DSB optical modulators can be employed to generate symmetrical waveforms. If asymmetrical waveforms are desired, single sideband modulation (SSB) is necessary. Figure 2 shows the spectra of a two-tone signal under different modulation formats and their corresponding output time-domain waveforms in a period, where the modulations of common DSB, DSB with carrier suppression and SSB with carrier suppression are assumed. In this figure, the real-time Fourier transform in the temporal Talbot system is seen, where the spectrum of the modulated signal is mapped onto the output time-domain waveform.

Fig. 2. The spectra (a), (c), and (e) of a two-tone signal under different modulation formats and their corresponding output waveforms (b), (d), and (f) in a period. Common DSB modulation is shown in (a) and (b); DSB modulation with carrier suppression is shown in (c) and (d); SSB modulation with carrier suppression is shown in (e) and (f). ${t_1} = |\ddot{\Phi }|{\omega _1}$ represents the distance between the desired position and the center position in each cycle, so as ${t_2} = |\ddot{\Phi }|{\omega _2}$.

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3. Proof-of-concept experiment

A proof-of-concept experiment has been implemented. The experimental setup is shown in Fig. 3. The setup consists of two parts; i.e., the pulse generator and the real-time Fourier transformer based on the temporal Talbot effect. In the pulse generator [22–23], a continuous wave (CW) light at 1550 nm with an optical power of 10 dBm is launched into two cascaded Mach-Zehnder modulators (MZMs). A microwave signal with a frequency of 9.38 GHz from a RF source (R&S SMB 100A) is fed to two MZMs after the RF signal is split and each signal is amplified by an RF amplifier (TRSPA-001300G25). A tunable optical delay line (General Photonics VDL-001) is placed between the two MZMs to compensate for the phase difference between the two RF links. The first MZM is biased at the minimum transmission point and the second one is biased at the quadrature point. In the Fourier transformer, the repetitive pulse train from the pulse generator is first amplified by a booster erbium-doped fiber amplifier (EDFA; Amonics AEDFA-23-B-FA) and then is modulated by a RF signal under test from a second RF source. The modulated pulse train propagates through two spools of dispersion compensation fiber (DCF). An in-line EDFA (Conquer KG-EDFA-P) is placed between the two spools of fiber to compensate for the power loss. The total dispersion amount of the two spools of DCF is measured to be 1808.90 ps² by using the measurement method given in [24]. Note that the pulse repetition period T and the dispersion amount $\ddot{\Phi }$ satisfy the Talbot condition $\textrm{|}\ddot{\Phi }\textrm{|= }{T^2}/2\pi $. The optical signal after the dispersion is detected by a photodetector (PD). The detected signal is then recorded by a sampling oscilloscope (Anritsu MP2100B), which is triggered by the signal from the first RF source. Though there is no requirement on synchronization between the pulses and the injected RF signal (RF 2), the sampling oscilloscope needs to be triggered by the RF signal (RF 1) for generating the pulse train in order to record the generated waveforms.

Fig. 3. Schematic diagram of the experimental setup. CW: continuous wave, MZM: Mach-Zehnder modulator, TODL: tunable optical delay line, EDFA: Erbium-doped fiber amplifier, DCF: dispersion compensation fiber, PD: photodetector, OSC: sampling oscilloscope, RFA: radio frequency amplifier.

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The recorded pulse train from the pulse generator without any modulation and dispersive propagation is shown in Fig. 4(a). The pulse train without RF modulation after the dispersion is shown in Fig. 4(b). It is seen that the period and profile of the pulse train almost keeps unchanged. Therefore, it is evident that the integer temporal Talbot phenomenon is observed. Next, a RF signal with a frequency of 4.69 GHz is modulated on the pulse train. The recorded waveforms under different injected RF power are shown in Fig. 5. As expected, new sub-pulses corresponding to the first-order sidebands appear, as compared with the case without any RF modulation. Moreover, the amplitude of the sideband pulses increases as the injected RF power increases. The time interval between the carrier and its corresponding sideband pulses is measured to be 53.3 ps, which perfectly matches the theoretical prediction ${T_1} = |\ddot{\Phi }|\omega $. It should be noted that, since the recorded pulses are broadened due to the limited bandwidth (25 GHz) of the applied sampling oscilloscope in the experiment, we chose a frequency of the modulating RF signal such that the pulses corresponding to the upper- and low-sidebands completely overlap and are located in the middle of two adjacent carrier pulses. Nevertheless, the given experimental results clearly demonstrate the real-time Fourier transform based on the temporal Talbot effect and verify the correctness of the given theoretical results.

Fig. 4. The recorded pulse trains without RF modulation (a) before the dispersion and (b) after the dispersion. T₀=106.6 ps.

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Fig. 5. The recorded waveforms after the pulse train is modulated by a 4.69 GHz RF signal and propagates through the dispersion under different injected RF power, (a) −14d Bm, (b) −15 dBm, (c) −16 dBm, and (d) −17 dBm. T₁=53.3 ps.

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4. Simulation results

We present some numerical examples using a computer simulation program to demonstrate the proposed AWG method. In the simulation, it is assumed that each optical pulse is Gaussian-shaped as $\exp [ - {t^2}/(2\tau _0^2)]$ and the pulse width ${\tau _0}$ is set to be 3.3 ps. The period of the pulse train is set to be 205 ps. The repetition rate is therefore 4.9 GHz, which means the maximum frequency of the modulating signal is 2.45 GHz in order to avoid aliasing distortion. The dispersion amount set according to the Talbot condition $\textrm{|}\ddot{\Phi }\textrm{|= }{T^2}/2\pi $ is 6738 ps². The modulator in the simulation is a push-pull MZM and the expression for DSB modulation is as $E(t) = \exp [jx(t) + \varphi ] + \exp [ - jx(t)]$, in which $\varphi $ is the bias phase shift, $x(t) = \sum\limits_i {{\alpha _i}\cos ({\omega _i}t)} $ and ${\alpha _i}$ is the modulation index. In the first example, we aim to generate a pulse train with a triangle envelope. Due to the symmetry of the desired waveform, the DSB modulation with carrier is employed. We use a multi-tone signal with eight frequencies to modulate the input pulse train. The frequencies are 0.4050 GHz, 0.8100 GHz, 1.2150 GHz, 1.6200 GHz and 2.0250 GHz, and the corresponding modulation indices are 0.9129, 0.8165, 0.7071, 0.5774, and 0.4082 respectively. The power ratio of the multiple tones is as 1.000:0.8002:0.5998:0.3997:0.2000. The spectrum of the modulated signal, the output temporal waveform and the electrical spectrum of the extracted envelope are shown in Figs. 6(a)–6(c), respectively. It is seen a perfect pulse train with an envelope of triangle waveform is generated. The simulation result matches the result predicted by the theoretical model very well, which also indicates the correctness of the given theoretical model. After O/E conversion, a low-pass filter with a cut-off frequency of about 33.5 GHz can be utilized to extract the envelope from the pulse train, which is also shown in Fig. 6. There are slight ripples in the extracted waveform, which can be owned to the non-ideal bandlimited interpolation in the low-pass filtering.

Fig. 6. (a) The spectrum of the modulated multi-tone signal (DSB). (b) The generated pulse train (red) with a triangle waveform as well as the extracted envelope (blue) after low-pass filtering. (c) The electrical spectrum of the extracted envelope.

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In the second example, we aim to generate a pulse train with a saw-tooth envelope. Since the target waveform is asymmetric, the SSB modulation with carrier suppression (SSB-CS) is employed in this case. Here we use a multi-tone signal with 11 frequencies to modulate the pulse train. The frequencies are 0.4050 GHz, 0.8100 GHz, 1.2150 GHz, 1.6200 GHz, 2.0250 GHz, 2.4300 GHz, 2.8350 GHz, 3.2400 GHz, 3.6450 GHz, 4.0500 GHz, 4.4550 GHz and the corresponding modulation indices are 0.0449, 0.0635, 0.0777, 0.0898, 0.0840, 0.0777, 0.0710, 0.0635, 0.0550, 0.0449, and 0.0317 respectively. The power ratio of the multiple tones is 1.0000:2.0001:2.9947:4.0000:3.5000:2.9947:2.5004:2.0001:1.5004:1.0000:0.4985. The spectrum of the modulated signal, the output temporal waveform and the electrical spectrum of the extracted envelope are shown in Figs. 7(a)–7(c), respectively. The extracted envelope waveform after the low-pass filtering is also given. Again, the simulation result agrees with the theoretical prediction very well.

Fig. 7. (a) The spectrum of the modulated multi-tone signal (SSB-CS). (b) The generated pulse train (red) with a saw-tooth waveform as well as the extracted envelope (blue) after low-pass filtering. (c) The electrical spectrum of the extracted envelope.

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The proposed approach to generating a pulse train with arbitrary waveform has two major advantages. On one hand, it can increase the pulse number while keeping the envelope of each pulse unchanged by simply using a multi-tone signal. For a given number N of the sinusoidal signals, the pulse number is increased by a factor of 2N+1 if the DSB modulation is considered. On the other hand, we can use a signal with a relatively low frequency (or maximum frequency) to generate a waveform with a higher frequency. Due to the limit of the Nyquist sampling theorem, the maximum frequency of the signal modulating on the pulse train is less than the half of the pulse repetition rate, since the temporal position of the output pulse is related to the modulating frequency as $\omega = t/\ddot{\Phi }$. If the frequency is greater than half of the frequency of the input pulse train, the corresponding output pulse will appear in adjacent periods. Due to the increase in the pulse number of the generated pulse train, the maximum frequency of the generated pulse envelope is increased accordingly, which can be clearly seen from the above two examples. Figure 8 shows another example in which the output pulse train is enveloped with a sinusoidal waveform. The frequencies in the modulating signal are 0.4050 GHz, 0.8100 GHz, 1.2150 GHz, 1.6200 GHz, and 2.0250 GHz, and the corresponding modulation indices (SSB-CS modulation) are 0.1111, 0.1924, 0.2221, 0.1924 and 0.1111, respectively. The power ratio of the multiple tones are 1.0000:2.9990:3.9964:2.9990:1.0000. The frequency of the generated sinusoidal envelope is 10 GHz. It is evident that the frequency is increased by 4 times compared with the maximum frequency of the input analog signal.

Fig. 8. (a) The spectrum of the modulated multi-tone signal (SSB-CS). (b) The generated pulse train (red) with a sinusoidal waveform as well as the extracted envelope (blue) after low-pass filtering. (c) The electrical spectrum of the extracted envelope.

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For the proof of concept, two more patterns are generated to show the potential of the proposed scheme. We also generate the pulse trains with trapezoid and step-like envelopes as in [12]. Figures 9(a) and 9(b) show the simulation results of the generated trapezoid and step-like pulse trains, respectively. Here we use a multi-tone signal with 10 frequencies based on CSB modulation to generate the trapezoid pulse train. The frequencies are 0.4050 GHz, 0.8100 GHz, 1.2150 GHz, 1.6200 GHz, 2.0250 GHz, 2.4300 GHz, 2.8350 GHz, 3.2400 GHz, 3.6450 GHz, 4.0500 GHz and the corresponding modulation indices are 0.410, 0.415, 0.420, 0.425, 0.430, 0.435, 0.440, 0.445, 0.450, and 0.455, respectively. For the generation of the step-like envelope pulse train, the frequencies in the modulating signal are 0.4050 GHz, 0.8100 GHz, 1.2150 GHz, 1.6200 GHz, 2.0250 GHz, 2.4300 GHz, 2.8350 GHz and the corresponding modulation indices are 0.26, 0.26, 0.32, 0.32, 0.32, 0.26 and 0.26 respectively.

Fig. 9. Simulation results. (a) The generated pulse train with a trapezoid envelope and (b) the generated pulse train with a step-like envelope.

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It is interesting to compare the given AWG approach with the approach based on temporal pulse shaping (TPS). In a TPS system [9–11], the output optical signal is also the scaled Fourier transform of the modulated pulse train, which is the same as the presented work. However, in a TPS system, a pair of dispersion elements with conjugate dispersion values should be placed before and after an EOM, via which an electrical RF signal is modulated on a time-stretched pulse train. The dispersion mismatch or higher-order dispersion in the applied dispersion element pair would lead to the distortion in the generated optical waveform. In addition, the dispersion value in TPS can be varied, but should satisfy the far-field condition in order to realize the time-domain filtering [25]. It is worth comparing this work with the AWG approach based on the discrete Fourier transform (DFT) inherent in the Talbot effect [15–16]. Our work is based on the real-time Fourier transform on an input analog signal, which is a mapping process from the frequency domain (before dispersion) to the time domain (after dispersion). However, in the DFT-based AWG approach, the amplitudes of the optical pulses before and after dispersion are a DFT pair. The period of the modulating discrete signal should be integer multiples of the period of the pulse train in the DFT-based approach; while in our work, there is no such limitation on the period of the modulating analog signal.

5. Conclusion

In summary, a novel photonic approach for generating pulse train with arbitrary waveform has been proposed in this paper. The approach is based on the property of real-time Fourier transform in the temporal Talbot effect, where the spectrum of the modulating analog signal is converted into the envelope of the output pulse train in each period. A general and strict theoretical framework is presented for the first time, which shows that the Fourier transform relationship between the optical signals before and after the dispersion holds true given that the dispersion amount satisfies the Talbot condition, which has been fully approved by the given experimental and simulation results. We proposed to generate symmetrical or asymmetrical arbitrary waveform by using DSB or CSB modulation according to the symmetry in the spectrum of modulated optical signal. The major advantage of the given approach lies in that it can generate a repetition-rate multiplied optical pulse train with arbitrary waveform by simply using a multi-tone RF signal with appropriate frequencies and powers. Compared with the TPS-based approach, the presented scheme avoids the problem of dispersion match; compared with the DFT-based approach, the given scheme requires no synchronization between the pulse train and the modulation signal. We believe the proposed scheme is a potential solution to AWG with a relatively simple structure and with a low requirement on the input signal.

Funding

National Key Research and Development Program of China (2019YFB2203200); National Natural Science Foundation of China (61975048, 41905024, 61901148, 62001148); Natural Science Foundation of Zhejiang Province (LZ20F010003, LQ20F010008).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Photonic arbitrary waveform generation based on the temporal Talbot effect

Abstract

1. Introduction

2. Principle

3. Proof-of-concept experiment

4. Simulation results

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (9)

Optics Express