Real-Time Reconfigurable Radio Frequency Arbitrary-Waveform Generation via Temporal Pulse Shaping with a DPMZM and Multi-Tone Inputs

Abstract

Benefitting from a large bandwidth and compact configuration, a time-domain pulse-shaping (TPS) system provides possibilities for generating broadband radio frequency (RF) arbitrary waveforms based on the Fourier transform relationship between the input–output waveform pair. However, limited by the relatively low sampling rate and bit resolution of an electronic arbitrary-waveform generator (EAWG), the diversity and fidelity of the output waveform as well as its reconfiguration rate are constrained. To remove the EAWG’s limitation and realize dynamic real-time reconfiguration of RF waveforms, we propose and demonstrate a novel approach of RF arbitrary-waveform generation based on an improved TPS system with an integrated dual parallel Mach–Zehnder modulator (DPMZM) and multi-tone inputs. By appropriately adjusting the DC bias voltages of DPMZM and the power values, as well as the center frequencies of the multi-tone inputs, any desired RF arbitrary waveform can be generated and reconfigured in real time. Proof-of-concept experiments on the generation of different user-defined waveforms with a sampling rate up to 27 GSa/s have been successfully carried out. Furthermore, the impact of modulation modes and higher-order dispersion on waveform fidelity is also discussed in detail.

1. Introduction

Broadband radio frequency (RF) arbitrary-waveform generation (AWG) is of great significance in modern information systems, including high-speed optical communications [1], biomedical imaging [2], chemical coherence control [3], and advanced radar applications [4]. The ultrashort optical pulse shaping, which is always adopted for RF AWG by using spatial or temporal signal-processing techniques, attracts a great deal of research attention due to its ultrafast waveform synthesis capability with response time in the scale of femtoseconds [5,6]. The spatial optical pulse shaping scheme was initially proposed and experimentally demonstrated by A. M. Weiner et al. [7]. The incident optical pulse is firstly decomposed in spectrum by a diffraction grating to achieve frequency-to-space linear mapping. And then be modulated in spatial amplitude or phase by using either fixed-phase masks or a programmable spatial light modulator (SLM) to achieve the desired spectral shaping. The spectral tailored pulses are coupled into a subsequent fiber connected with a photodetector (PD) to obtain the desired RF arbitrary waveforms. Major drawbacks of this spatial optical pulse-shaping system lie in its bulky and costly configuration, high insertion loss, calibration difficulty, and limited integration capability. In addition, the bandwidth of typically utilized pulse shapers, such as crystal SLM is limited into tens of KHz, which restrict the configuration rate of the output RF waveform to millisecond time scale [8]. As the time-domain equivalent of spatial pulse shaping, temporal pulse shaping (TPS), where spatial dispersion is replaced by all-fiber temporal dispersion and the SLM is used instead of a broadband electro-optical modulator, is widely investigated and adopted for RF AWG [9,10,11,12,13,14,15,16,17,18]. Benefitting from the huge bandwidth offered by electro-optic modulator and the low loss of all-fiber structure as well as the compact system configuration, the TPS can significantly improve the system stability and the configuration rate of the output waveforms in contrast to the spatial optical pulse-shaping scheme. In Ref. [10], the update rates of the generated RF waveforms can be increased into sub-GHz range. However, the essence of the TPS scheme for RF AWG is the Fourier transform relation between the input–output waveform pair. To achieve the desired RF waveform, the electronic arbitrary waveform generator (EAWG) is required to generate the corresponding input signal, which greatly limits the output waveform diversity and fidelity due to the relative low sampling rate and bit resolution of EAWG, i.e., high-fidelity square or sawtooth waveforms are hard to be obtained since high-resolution broadband Sinc or Sinc2 input signals are difficult to be generated by current commercial EAWGs.

During past decades, much research has been devoted to improving the TPS system. To overcome the limitation of the Fourier transform relation between the input–output waveform pair, an unbalanced TPS system for RF AWG was proposed in Ref. [11], in which two dispersive elements have opposite chromatic dispersion but not identical in magnitude. The entire system can be considered as a balanced TPS system with the residual dispersion, which lead to the generated waveform with a temporal shape as a scaled version of the input signal. However, the unbalanced TPS system still require EAWG to provide the input RF signal. Another TPS system reported in Ref. [15] utilizes phase modulation (PM) instead of AM to improve the system performance in terms of the waveform diversity. Moreover, the required input signal can be provided by the high-rate pulse pattern generator (PPG), which removes the requirement of the EAWG. However, this approach requires iterative algorithms to improve the waveform fidelity by applying the optimal phase information. The time-consuming iterative algorithms significantly restrict the waveform reconfigurability, i.e., the desired RF waveforms cannot be generated in real-time. Besides the RF AWG, the TPS system with amplitude modulation (AM) also can function as a high-bandwidth microwave spectrum analyzer [16]. Driven by the RF signal with multiple frequency elements, the system creates copies of the original pulse at temporal shifts corresponding to the spectral spacing of the input signal. Based on this function, a TPS system with multi-tone inputs was proposed and demonstrated in our previous work to realize the RF AWG with simple manipulation and high stability [18]. By adjusting the center frequency and power values of sinusoidal generators instead of EAWG, a list of delayed and weighted optical pulses serving as the samples for the desired RF waveform can be generated. However, our previous work employs a push–pull Mach–Zehnder modulator (MZM) to realize the intensity modulation. Each frequency element of the RF input is mapped into two symmetrical output pulses with the time interval between them depending on the frequency value. Accordingly, the RF waveform generated in our previous work is limited to be symmetrical but not arbitrary.

In this paper, we propose a novel real-time reconfigurable RF AWG method based on an improved temporal pulse-shaping system with an integrated dual-parallel MZM (DPMZM) and multi-tone RF inputs. In this approach, by properly adjusting the DC bias voltages of DPMZM, the flexible modulation modes, including double-sideband (DSB) modulation, single sideband (SSB), and carrier-suppressed single sideband modulation (CS-SSB), of the multi-tone inputs can be achieved. Thanks to the variable modulation types offered by DPMZM, any RF waveform with self-defined temporal profile can be obtained or reconfigured in real-time by simply adjusting the power values and the frequencies of the multi-tone inputs. Proof-of-concept experiments on the generation of different waveforms with a sampling rate up to 27 GSa/s are successfully carried out. In addition, the influences of different modulation modes and higher-order dispersion on system performance in terms of waveform diversity and fidelity are also discussed in detail.

2. Principle of Operation

The schematic diagram of the proposed real-time reconfigurable RF AWG system via TPS with a compact DPMZM and multi-tone inputs is illustrated in Figure 1. Different from the conventional TPS system, the proposed system is built using a mode-locked laser (MLL), a DPMZM, and a pair of conjugate dispersive mediums (DM1 and DM2). One major difference lies in the employment of DPMZM, which could realize the CS-SSB modulation mode to guarantee the one-to-one mapping from each frequency element to the output optical pulse. Another difference is the generation of the multi-tone RF inputs, which is offered by commercial sinusoidal signal generators instead of EAWG. Based on the Fourier transform relationship between the RF input signal and the output optical waveform, the spectrum of the applied multi-tone signal is linearly mapped into the temporal profile of the output optical signal, i.e., a list of discrete optical pulses with adjustable amplitudes and time intervals are generated. The following PD and low-pass filter (LPF) detect and smooth the temporal envelope of optical pulses to realize the RF waveform generation. Note that the output optical pulses serve as the sampling points of the desired RF waveforms. In this design, by appropriately presetting the DC bias voltages of the DPMZM, various modulation modes including DSB, SSB, and CS-SSB can be switched to meet different application requirements. Additionally, by simply configuring the frequency spacing and amplitudes of the multi-tone RF inputs, the desired RF arbitrary waveform can be generated and reconfigured in real-time.

The ultra-short optical pulse emitted from the MLL is assumed as transform-limited. The electric field of each pulse is denoted by $E_1\left(t\right)=\exp \left(-t^2/2{\tau }_0^2\right)$, and its spectrum is $E_1\left(\omega \right)=\mathcal{F}\left\{E_1\left(t\right)\right\}=\sqrt{2\pi }{\tau }_0\exp \left(-{\tau }_0^2{\omega }^2/2\right)$, where ${\tau }_0$ denotes the pulse width at $1/e$ peak intensity, and $\mathcal{F}\left\{\cdot \right\}$ indicates the Fourier transform. The optical pulse from the MLL initially enters the first dispersion medium DM1 to be chirped in spectrum and broadened in the time domain. We consider DM1 consisting of a coil of single-mode fiber (SMF) with a dispersion value of $\ddot{\mathsf{\Phi }}$. The dispersion value is defined as the first-order derivative of the group delay with respect to angular frequency [19]. Meanwhile, the signal loss and the frequency-independent group delay induced by DM1 are ignored. The frequency transfer function of DM1 can be expressed as $H_1\left(\omega \right)=\exp \left(j\ddot{\mathsf{\Phi }}{\omega }^2/2\right)$. When the far-field condition is satisfied, i.e., ${\tau }_0^2/\ddot{\mathsf{\Phi }}<<1$ [20], the optical pulse out of DM1 will be broadened, with its envelope being the time-scaled Fourier transform of the initial input $E_1\left(t\right)$. Hence, the frequency-domain expression of the broadened pulse out of DM 1 can be written as $E_2\left(\omega \right)=E_1\left(\omega \right)H_1\left(\omega \right)$. And its time-domain expression can be expressed as $E_2\left(t\right)={\mathcal{F}}^{-1}\left\{E_2\left(\omega \right)\right\}$, where ${\mathcal{F}}^{-1}\left\{\cdot \right\}$ denotes the inverse Fourier transform.

Then, the output optical pulses of DM1 are injected into DPMZM to be modulated by the input RF signal, which consists of multiple sinusoidal signals with equal frequency spacing. Assume that each frequency element has a corresponding amplitude value $A_n\left(1\le n\le N\right)$ and center frequency $f_n\left(1\le n\le N\right)$, where $n$ is an integer. To realize the CS-SSB modulation, the RF signal is split into two parts and separately applied into two arms of DPMZM through a 90-degree phase shifter. The RF signals applied into two arms of the DPMZM can be expressed as follows:

$$\begin{array}{l}{x}_1\left(t\right)={\displaystyle \sum _{n=1}^N{A}_n\cos \left({\omega }_{n}t\right)}\\ x_2\left(t\right)={\displaystyle \sum _{n=1}^N{A}_n\cos \left({\omega }_{n}t+\pi /2\right)}\end{array}$$

(1)

where ${\omega }_n=2\pi f_n\left(1\le n\le N\right)$. Meanwhile, both sub-modulators (MZM1 and MZM2) of the DPMZM operate in push–pull mode, with the DC bias voltages of the upper and lower arms set at the minimum transmission point, i.e., $V_{B1}$ and $V_{B2}$ is set to $V_{\pi }$. And the main MZM works at the quadrature bias point, i.e., $V_{B3}$ is set as $V_{\pi /2}$; the output optical signal from the DPMZM can be written as follows:

$$\begin{array}{l}{E}_3\left(t\right)={\displaystyle \frac{1}{2}}{E}_2\left(t\right)\cdot {\displaystyle \sum _{n=1}^N{A}_n\left[\cos \left({\omega }_{n}t\right)\exp \left(j\pi /2\right)+\cos \left({\omega }_{n}t+\pi /2\right)\right]}\\ \hspace{1em}\hspace{1em}\hspace{0.17em}={\displaystyle \frac{1}{2}}j{E}_2\left(t\right)\cdot {\displaystyle \sum _{n=1}^N{A}_n\exp \left(j{\omega }_{n}t\right)}\end{array}$$

(2)

The Fourier transform of $E_3\left(t\right)$ is written as follows:

$$E_3\left(\omega \right)={\displaystyle \frac{1}{2}}j{E}_2\left(\omega \right)\ast {\displaystyle \sum _{n=1}^N{A}_n\delta \left(\omega -{\omega }_n\right)}={\displaystyle \frac{1}{2}}j{\displaystyle \sum _{n=1}^N{A}_n{E}_2\left(\omega -{\omega }_n\right)}$$

(3)

where $\ast$ denotes the convolution calculation.

Next, the modulated optical signal is injected into DM2 with the conjugated dispersion to compensate for the dispersion introduced by DM1 to achieve the desired discrete pulse train. The transfer function of DM2 can be expressed as $H_2\left(\omega \right)=\exp \left(-j\ddot{\mathsf{\Phi }}{\omega }^2/2\right)$, which has the opposite dispersive effect of DM1. Therefore, the frequency-domain expression of optical signal after DM2 is written as follows:

$$\begin{array}{l}{E}_4\left(\omega \right)=E_3\left(\omega \right)H_2\left(\omega \right)\\ \hspace{1em}\hspace{1em}\hspace{0.33em}\hspace{0.33em}\hspace{0.17em}={\displaystyle \frac{1}{2}}j{\displaystyle \sum _{n=1}^N{A}_n{E}_1\left(\omega -{\omega }_n\right)H_1\left(\omega -{\omega }_n\right)H_2\left(\omega \right)}\\ \hspace{1em}\hspace{1em}\hspace{0.33em}\hspace{0.33em}\hspace{0.17em}={\displaystyle \frac{1}{2}}j{\displaystyle \sum _{n=1}^N{A}_n{E}_1\left(\omega -{\omega }_n\right)\exp \left(-j\ddot{\mathsf{\Phi }}\omega {\omega }_n\right)\exp \left(j\ddot{\mathsf{\Phi }}{\omega }^2/2\right)}\end{array}$$

(4)

Based on inverse Fourier transform, the time-domain expression of $E_4\left(t\right)$ can be written as follows:

$$E_4\left(t\right)={\mathcal{F}}^{-1}\left\{E_4\left(\omega \right)\right\}={\displaystyle \frac{1}{2}}j{\displaystyle \sum _{n=1}^N{A}_n{E}_1\left(t+\ddot{\mathsf{\Phi }}{\omega }_n\right)\exp \left[j{\omega }_n\left(t-\ddot{\mathsf{\Phi }}{\omega }_n/2\right)\right]}$$

(5)

The following PD performs envelope-detection on the output optical pulses. According to the principle of square-law detection, the current output of the PD can be represented as follows:

$$i\left(t\right)\propto {\left|E_4\left(t\right)\right|}^2\approx {\displaystyle \frac{1}{4}}{\displaystyle \sum _{n=1}^N{A}_n^2{\left|E_1\left(t+\ddot{\mathsf{\Phi }}{\omega }_n\right)\right|}^2}$$

(6)

As can be seen, the detected current consists of only one term, representing the overall energy of the up-shifted sidebands of the modulated multi-tone signal. Meanwhile, the linear one-to-one mapping from frequency elements to output pulses serving as the sampling points of output waveform is realized. Moreover, the output waveform depends on the amplitude $A_n\left(1\le n\le N\right)$ and center frequency $f_n\left(1\le n\le N\right)$ of the input multi-tone signal. By adjusting these two values appropriately, any desired temporal profile of the output pulse train or the RF waveform can be achieved and reconfigured in real-time. Meanwhile, the sampling interval or the time interval $\Delta \tau$ of adjacent sampling points is dominated by both the system dispersion value $\ddot{\mathsf{\Phi }}$ and the frequency spacing $\Delta f$ of the input multi-tone signal. Hence, we can obtain the sampling interval of the generated waveform by following:

$$\Delta \tau =\ddot{\mathsf{\Phi }}2\pi \left(f_{n+1}-f_n\right)=\ddot{\mathsf{\Phi }}2\pi \Delta f$$

(7)

The major merit of the designed system with an integrated DPMZM lies in that flexible modulation modes can be achieved by simply adjusting the DC bias voltages. Thus, the CS-SSB, DSB, and SSB modulation modes, which correspond to the symmetrical and asymmetrical output waveforms, can be obtained. By simply presetting the DC biases as $V_{B1}=V_{B2}=V_{\pi /2}$, and applying the multi-tone input into sub-MZM1 directly, the DSB modulation of the input signal on the chirped optical pulses can be achieved and the system output current can be expressed as follows:

$$i_{\mathrm{DSB}}\left(t\right)\propto {\left|E_1\left(t\right)\right|}^2+{\displaystyle \frac{1}{4}}{\displaystyle \sum _{n=1}^N{A}_n^2{\left|E_1\left(t-\ddot{\mathsf{\Phi }}{\omega }_n\right)\right|}^2}+{\displaystyle \frac{1}{4}}{\displaystyle \sum _{n=1}^N{A}_n^2{\left|E_1\left(t+\ddot{\mathsf{\Phi }}{\omega }_n\right)\right|}^2}$$

(8)

As shown in Equation (8), each frequency element of the RF input is mapped into two symmetrical sampling points with adjustable amplitudes and time intervals. Moreover, the central sampling point corresponding to the optical carrier also exists to guarantee the symmetry of the output RF waveforms. Noting that making an appropriate adjustment to the main bias $V_{B3}$ can effectively suppress the carrier wave to a certain extent.

Similarly, by adjusting the DC bias voltages as $V_{B1}=V_{\pi /2}$, $V_{B2}=V_{\pi /2}$, $V_{B3}=V_{\pi /2}$, and applying the multi-tone input into two sub-MZMs with a 90-degree phase shift, the SSB modulation mode corresponding to the asymmetrical RF waveform generation can be obtained. And the output current can be obtained as follows:

$$i_{\mathrm{SSB}}\left(t\right)\propto {\left|E_1\left(t\right)\right|}^2+{\displaystyle \frac{1}{4}}{\displaystyle \sum _{n=1}^N{A}_n^2{\left|E_1\left(t+\ddot{\mathsf{\Phi }}{\omega }_n\right)\right|}^2}$$

(9)

As can be seen from Equation (9), the detected current consists of the overall energy of up-shifted sidebands and that of optical carrier, which leads to the pulse train with asymmetrical temporal profile.

Finally, the detected pulse train is smoothed by a LPF to obtain the desired RF waveform. By simply adjusting the amplitudes and frequency values as well as the DC bias voltages of DPMZM, any self-defined RF waveform can be achieved.

3. Experiment and Discussions

3.1. Arbitrary Signal Generation via TPS with CS-SSB Modulation

A proof-of-concept experiment with the setup as shown in Figure 2 was carried out. An MLL (Menlo ELMO 780, MenloSystems, Germany) is employed to generate the ultra-short pulse train with a repetition period of 10 ns and a spectrum range about 40 nm centered at 1550 nm. The pulse train, with each pulse duration of approximately 90 fs, is directly sent to SMF, which works as the dispersive medium of DM1. The length and dispersion value of SMF are 62.54 km and 1064.8 ps/nm, respectively. Then, the chirped optical signal time broadened by SMF is delivered into the DPMZM (MXIQER-LN-30) through the polarization controller (PC), which is used to optimize the polarization state of the input optical signal to maximize the output optical power of the DPMZM. The modulator has a RF-operating bandwidth of 40 GHz, and its half-wave voltage is about 7 V. In order to realize CS-SSB modulation, the DC biases of the two sub-MZMs of the DPMZM are both set at the minimum point (i.e., $V_{B1}=V_{B2}=7 \mathrm{V}$), and the DC bias of the main MZM is set at the quadrature point, i.e., $V_{B3}=4.5 \mathrm{V}$. Limited by our current lab conditions, the signal generation module (SGM) is built by three RF signal generators (R&S SMB100A, Ceyear 1435F, R&S ZNB40), two electric couplers (ECs), and two RF amplifiers (Qotana DBLNA601004000A, Chengdu, Sichuan, China), which could provide the adjustable three-tone RF inputs. By using a 90-degree phase shifter, the three-tone signal offered by SGM is split into two parts with a phase difference of $\pi /2$. Then, separately drive the two sub-MZMs of DPMZM to modulate the chirped optical pulse train. Next, the modulated optical signal is amplified by an EDFA (AEDFA-23-B-FA) to compensate the power loss from modulation and transmission. The output of EDFA is then delivered into the subsequent dispersion compensation module (DCM, YOSC DCM-SM-C100%-120-LCP-01), which works as the dispersion medium of DM2. Here, the DCM comprises two coils of DCF with a total length of 65 km and a combined dispersion value of −1064.8 ps/nm. Finally, a PD (Optilab PD-40-M) with a 40 GHz bandwidth detects the optical pulse train output from the DCM. The output of PD is captured by a sampling oscilloscope (DSO, Anritsu MP2100B) with a 50 GHz bandwidth, which should be synchronized with the MLL.

Firstly, we perform system dispersion matching to ensure the fidelity of the output RF waveform. The optical pulses before (i.e., emitted from MLL) and after dispersion matching (i.e., through conjugate dispersion but without RF inputs) are shown in Figure 3. Obviously, the output pulse after dispersion matching fits well with the original case, and the little distortion mainly comes from the higher-order dispersion of the system, which will be discussed in detail.

Subsequently, we investigate the RF AWG capability of our proposed approach by adjusting the amplitudes or frequencies of multi-tone inputs from the SGM. Limited by the RF signal generator (SG) number and the bandwidth of the utilized ECs, the SGM is built by three SGs, and its offered maximum frequency is not larger than 12.5 GHz. By presetting the frequency values of three-tone RF inputs as 2.5 GHz, 7.5 GHz, and 12.5 GHz, the achieved output pulses serving as the sampling points of the desired waveforms are given in Figure 4a–d. Among these, the blue solid points represent the results captured by the sampling oscilloscope, while the red dashed lines display the fitting temporal profiles of the generated waveforms. Obviously, three output pulses corresponding to the three-tone RF signal are generated, and the time interval between any two adjacent points is approximately 46 ps. Since the frequency interval of the multi-tone inputs is 5 GHz and the system dispersion amount is about 1068.6 ps/nm, the theoretical time interval derived based on Equation (8) is about 43 ps. The measured time interval or sampling interval fits well with the theoretical value. Thus, our designed system can generate RF arbitrary waveform with a sampling rate exceeding 20 GSa/s, which allows for a maximum frequency exceeding 10 GHz according to the Nyquist sampling theorem. Note that the detected sampling pulses are broadened in the time domain because of the 40 GHz bandwidth of PD, which corresponds to the time resolution of 25 ps. By properly presetting the powers of three-tone input as 8 dBm, 11 dBm, and 10 dBm, the captured pulses with self-defined amplitude values are obtained and given in Figure 4a. Accordingly, the desired RF waveform can be easily obtained by smoothing the envelope of the recorded pulse train. Compared with the traditional TPS system, the proposed approach perfectly removes the limitation of EAWG, adopts DPMZM to realize the CS-SSB modulation, and utilizes the multi-tone inputs from the SGM to obtain the desired RF waveforms. Moreover, by adjusting the amplitudes and frequency values of the multi-tone inputs, the self-defined RF waveform with adjustable sampling rate can be generated. To further investigate the reconfiguration of the proposed system, the powers of three-tone RF inputs are, respectively, adjusted to 9 dBm, 8 dBm, and 9 dBm; the captured pulses are given in Figure 4b. Similarly, with the same frequency elements, the powers of three frequency elements are adjusted to 12 dBm, 10 dBm, 8 dBm or 8 dBm, 8 dBm, and 8 dBm; the measured pulses are presented in Figure 4c,d. As can be seen, by adjusting the amplitudes or the frequencies of the multi-tone inputs, the generated RF waveforms can be reconfigured dynamically. Note that the generated RF waveform is built by a list of discrete output pulse trains. Accordingly, the reconfiguration rate of the output waveform is kept same as its sampling rate. Based on Equation (7), the time interval or the sampling rate of the output RF waveform is dominated by system dispersion amount and the frequency spacing of multi-tone inputs. Hence, for the fixed system conjugate dispersion, the bandwidth of the RF waveform can be reconfigured in real-time by simply adjusting the frequency spacing of the multi-tone inputs. Theoretically, a smaller frequency spacing will facilitate the generation of RF waveform with a higher sampling rate or waveform bandwidth.

The major merit of DPMZM lies in that the flexible modulation mode can be obtained by simply adjusting its DC bias voltages. Different modulation modes lead to different types of output waveforms. Subsequently, we also experimentally demonstrate the RF waveform generation based on our designed system with DSB and SSB modulation modes.

3.2. Symmetric Waveform Generation via TPS with DSB Modulation

To achieve DSB modulation, one sub-MZM of the DPMZM is biased at the quadrature point and driven with the RF input, while the other sub-MZM remains idle. By appropriately adjusting the DC bias voltages of the idle sub-MZM and the main MZM, partial suppression of the optical carrier can be achieved. This suppression alleviates the limitations of excessive energy from optical carriers and enhances the diversity of the output waveforms. Here, we adjust the SGM into a single sinusoidal generator. By adjusting the frequencies or the powers of RF inputs, the symmetrical pulse sequence or RF waveforms can be obtained under DSB modulation. The measured waveforms including square or triangular shapes are given in Figure 5. By presetting the DC biases as 9 V, the frequency of RF input as 4 GHz, and its power value as 18.8 dBm, the captured output pulses are shown in Figure 5a. The duration time of the generated waveform is about 74 ps, and its time interval is 37 ps, corresponding to the sampling rate of 27 GSa/s. To reconfigure the output square waveform, the frequency of the input RF signal is adjusted into 8 GHz; the obtained outputs are given in Figure 5b. Obviously, the waveform duration is increased from 74 ps to 147 ps as the frequency of input signal increases. Furthermore, we change the frequency of RF input into 7 GHz and preset its power value as 8.9 dBm. The captured output pulses with a triangular envelope are displayed in Figure 5c. To edit the temporal envelope of the generated triangular waveform, the power value of the RF input is reset as 10 dBm while keeping its frequency same as 7 GHz. The obtained results are presented in Figure 5d. As can be seen, the output waveform duration stays the same, 121 ps, under the same input frequency, and its amplitudes increase as the input power grows up. Note that a single frequency element in the RF input leads to two symmetrical output pulses around the middle one from optical carrier, which guarantee the symmetry characteristic of the output RF waveform.

3.3. Asymmetric Waveform Generation via TPS with SSB Modulation

To achieve single sideband modulation, both sub-MZMs and the DPMZM are biased at the quadrature point, i.e., $V_{B1}=V_{B2}=3.5 \mathrm{V}$, $V_{B3}=4.5 \mathrm{V}$. By using a 90-degree phase shifter, the RF signal is split into two parts and separately drives two sub-MZMs with a 90-degree phase difference. Here, we change the frequency elements offered by SGM from one to three. The recorded results are given in Figure 6. Firstly, we use a 4 GHz single-tone signal as the input, two discrete pulses, respectively, corresponding to the input signal and the optical carrier can be generated, as shown in Figure 6a. The time interval or the sampling interval is about 38 ps, which leads to the square waveform with a bandwidth close to 26 GHz. Then, we adjust the input signal offered by SGM as a two-tone input, and their frequencies are 6 GHz and 12 GHz. The measured output pulses and their fitting sawtooth waveform are given in Figure 6b. To increase the sampling points inside one waveform period, we change the input signal offered by SGM into a three-tone input, and their frequencies are set as 4 GHz, 8 GHz, and 12 GHz. Their power values are, respectively, fixed as 12.9 dBm, 10 dBm, and 8dBm. The recorded results are presented in Figure 6c. In addition, to validate the reconfiguration of our designed system with SSB modulation, we adjust the power values of three-tone inputs as 12.6 dBm, 12.7 dBm, and 8 dBm while keeping the same frequencies, the captured pulses and their fitting waveform profile are given in Figure 6d.

3.4. Impact of Higher-Order Dispersion

In our designed system, a pair of conjugate dispersion mediums are required to generate the desired RF arbitrary waveform, which means that the absolute dispersion value of DM 1 should be equal to that of DM 2. Slight dispersion mismatch will lead to the output waveform distortion [13]. As discussed in Ref. [18], residual second-order dispersion from dispersion mismatch would cause pulse broadening in the output, subsequently decreasing the fidelity and the bandwidth of the output waveform. Besides the dispersion mismatch, the system performance is also affected by the higher-order dispersion, which is often disregarded in numerous applications. However, the impact of higher-order dispersion, especially that of the third-order dispersion, will become pronounced because of its accumulation over long distances. Considering third-order dispersion, the frequency response of two dispersion modules can be written as follows:

$$\begin{array}{l}{H}_1\left(\omega \right)=\exp \left(j{\stackrel{..}{\mathsf{\Phi }}}_2{\omega }^2/2\right)\exp \left(-j{\stackrel{..}{\mathsf{\Phi }}}_3{\omega }^3/6\right)\\ H_2\left(\omega \right)=\exp \left(-j{\stackrel{..}{\mathsf{\Phi }}}_2{\omega }^2/2\right)\exp \left(j{\stackrel{..}{\mathsf{\Phi }}}_3{\omega }^3/6\right)\end{array}$$

(10)

where ${\stackrel{..}{\mathsf{\Phi }}}_2$ and ${\stackrel{..}{\mathsf{\Phi }}}_3$ denote second-order dispersion value and third-order dispersion value, respectively.

For the designed system with CS-SSB modulation, we assume that the input optical pulse has a time width of 100 fs, and the multi-tone input consists of frequencies from 3 GHz to 24 GHz with the frequency spacing of 3 GHz. All frequency elements have the equal power values. Figure 7 presents the output discrete pulses with (red line) and without considering the impact of third-order dispersion (blue line). Obviously, the impact of the third-order dispersion on different frequency components varies, mainly reflected in the variation in pulse amplitude. As can be seen, as the input frequency increases, the amplitude value of its corresponding output pulse decreases. Especially for the frequency spacing up to 21 GHz, the amplitude value will decrease more than 20%. Meanwhile, the pulse width also undergoes slight broadening as its amplitude decreases. Fortunately, for the proposed system, the decrease in amplitudes induced by both of higher-order dispersion and the unideal spectral response of employed EDFA can be compensated by adjusting the input power settings of the multi-tone inputs.

4. Conclusions

In summary, a novel approach of RF arbitrary-waveform generation via temporal pulse shaping (TPS) with an integrated dual parallel Mach–Zehnder modulator (DPMZM) and multi-tone inputs has been proposed and experimentally demonstrated. By properly adjusting the DC bias voltages of DPMZM, the carrier-suppressed single-sideband modulation of RF input can be achieved, which guarantees the one-to-one linear mapping from each frequency element of RF input to the output pulses. Any desired temporal profile of the output pulse train can be obtained by simply adjusting the frequencies or the amplitudes of the multi-tone inputs. Accordingly, the desired RF waveform is also obtained by detecting and smoothing the output discrete pulse train. Proof-of-concept experiments on different waveforms generation have been successfully carried out. Additionally, the impact of different modulation modes and higher-order dispersion on waveform diversity and fidelity are also investigated.