## Abstract

A new photonic scheme for various waveforms generation has been proposed and demonstrated. In the scheme, two cascaded single-drive LiNbO_{3} Mach-Zehnder modulators serve as pulse shaper and the polarization-dependent character of the modulators is fully exploited and utilized. By arranging the polarization states of the incident light, two different spectra are achieved on two orthogonal polarization components respectively. Finally, the desired waveforms can be obtained by superimposing the photocurrents of the two orthogonal signals on a photodetector. The detailed theoretical analyses and simulations are given. In the experiment, square-shaped waveform, triangular waveform and sawtooth (or reversed-sawtooth) waveform are obtained successfully. Furthermore, an approach to smoothing the sawtooth waveform with fewer harmonics is suggested and verified.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Radio frequency (RF) arbitrary waveform generation (AWG) has a wide range of applications in wireless communications, radar, measurement systems, and so on [1–3]. AWG using optical technology (optical arbitrary waveform generation, OAWG) provides new access to solving the electronic bottleneck problem in traditional electrical arbitrary waveform generation (EAWG) [3].

Currently, there are many methods to implement OAWG. One of the most popular ways for OAWG is Fourier synthesis. By accurately controlling the amplitudes and phases of optical spectral lines, arbitrary waveform generation can be achieved. Usually, a spatial configuration is employed to implement spectral line control [4], which makes a complex system. Meanwhile, the optical phase is easily affected by the surrounding perturbation. Frequency-to-time mapping (FTTM) is another way to achieve OAWG, in which the desired waveform can be generated by shaping the spectrum of optical pulses and mapping the spectral envelope to the time-domain [5,6]. However, the spectral lines are manipulated in groups, which reduces the flexibility and accuracy of waveform generation.

OAWG can also be achieved by means of time-domain processing [7,8]. In [7], Jiang et al. have confirmed that OAWG can be realized by carving continuous wave (CW) and overlapping the optical envelopes in time domain. In the works, the rectangular waveform, triangular waveform and half duty sawtooth waveform are generated successfully, and the full duty sawtooth waveform is obtained by means of multiplexing technology. The schemes show that the time-domain method is relatively straightforward and can avoid complex spectral line operation. However, in [7] two light sources are required, and the configuration is not compact.

In both Fourier synthesis and time-domain processing, OAWG based on CW modulation is widely used, since it is comparatively simple and efficient. It is reported that dual-parallel Mach-Zehnder modulator (DP-MZM), dual-drive Mach-Zehnder modulator (DD-MZM) and polarization modulator (Pol-M) have been used in many schemes to implement OAWG [9–14]. Recently, a scheme for triangular waveform generation, in which the polarization sensitivity of one single-drive LiNbO_{3} Mach-Zehnder modulator (MZM) is exploited, has been demonstrated [15]. Although this scheme has the advantages of simple configuration and lower cost, only triangular waveform is obtained.

In this paper, a simple and efficient scheme is proposed for diverse waveforms generation. Two LiNbO_{3} MZMs are cascaded as pulse shaper, and their polarization sensitivity is used to generate independent sidebands on two orthogonal polarization components of one light field. Simultaneously, the character that the orthogonal components do not beat is fully utilized. Then the desired RF waveform can be achieved by using the time-domain synthesis method. The theoretical analyses and simulations are established, and square-shaped waveform, triangular waveform and sawtooth (or reversed-sawtooth) waveform with the repetition frequency of 3 GHz are obtained experimentally. All the results show good performance.

## 2. Principle

The experimental setup is schematically illustrated in Fig. 1. A polarization controller (PC) placed in front of either LiNbO_{3} MZM is used to control the polarization state of the incident light. Due to the fact that MZM is a polarization-dependent device, only the polarization component of the incident light coinciding with the principal axis of the MZM can be modulated. Thus, different spectral properties can be achieved on two orthogonal components. Finally, the desired waveforms can be obtained through the superimposition of the photocurrents of the two orthogonal signals on a photodetector (PD). The polarizing beam splitter (PBS) is placed at the end to separate the signals in two orthogonal directions for observation.

Mathematically, the linearly polarized CW emitted by the laser diode (LD) can be described as

where ${E}_{\text{0}}$ is the optical field amplitude, and ${\omega}_{0}$ is the angular frequency of the optical field. Suppose the drive signals applied on the MZMs areAssuming there is an angle $\theta $ between the polarization direction of the incident light and the principal axis (X-axis) of MZM1, the two orthogonal components, ${E}_{1}^{X}(t)$ and ${E}_{1}^{Y}(t)$, of the output from MZM1 can be expressed as

Because only the harmonics with the same polarization state may contribute to beat, when the optical field is detected by a PD, the photocurrent is expressed as

In Eq. (7)

It can be perceived from the above equations that the desired harmonics with proper coefficients can be obtained by controlling the parameters of $\theta $, $\phi $, ${\varphi}_{i}$ and ${m}_{i}$.

#### 2.1 Square-shaped waveform

Just as described in [7], square-shaped waveform can be simply obtained through the first-stage modulation. The Fourier series expansion of a square-shaped waveform is given by

Comparing Eq. (10) and Eq. (9), the photocurrent may present a square-shaped envelope in time domain only if ${X}_{1}=-3{Z}_{1}$ and ${Y}_{1}\to 0$. Figure 2(a) shows the calculated values of ${X}_{1}$, ${Y}_{1}$ and ${Z}_{1}$. It can be found that square-shaped waveform can be obtained under the condition that ${m}_{1}=1.14\text{2}$. Figure 2(b) displays the corresponding simulation result.

#### 2.2 Triangular waveform

In principle, the Fourier expansion of a periodic triangular waveform is written as

It can be seen that the amplitude of higher order harmonics decreases sharply, and the sum of the first two harmonics can be a good approximation to the whole expansion. Thus, Eq. (11) is simplified as

According to Fig. 3, two sets of modulation sidebands are carried on the orthogonal components of the optical field respectively. Meanwhile, the photocurrent expressed by Eq. (6) can be rewritten as

When $\theta =\pi /\text{4}$, $\phi =\pi /\text{2}$, ${\varphi}_{\text{1}}\text{=}{\varphi}_{\text{2}}\text{=}\pi /\text{4}$ and ${m}_{1}={m}_{2}$, Eq. (13) is reduced to

Obviously, Eq. (14) can be equivalent to Eq. (12) only if ${X}_{2}=-9{Z}_{2}$. Figure 4(a) shows the calculated values of ${X}_{2}$ and ${Z}_{\text{2}}$ versus the modulation index. Once ${m}_{\text{1}}={m}_{\text{2}}=0.752$, the above condition can be satisfied. As a consequence, triangular waveform can be obtained at Output 2. Figure 4(b) displays the simulation result, in which the two photocurrents, ${i}_{1}$ and ${i}_{2}$ in Eq. (6) are superimposed for triangular waveform generation.

#### 2.3 Sawtooth waveform

The Fourier series expansion of a sawtooth waveform is given by

Considering the third-order approximation, the expansion is approximated by

According to Eq. (6), when $\phi =\pi /\text{4}$, the photocurrent can be obtained as follows.

Equation (17) can be regarded as the sum of the two terms, $Saw1$ and $Saw2$. Compared with Eq. (16), if the conditions that ${X}_{\text{1}}\text{=2}{Y}_{\text{2}}\text{=}-{\text{3Z}}_{1}$ and $S\text{aw}2\to 0$ are met, a sawtooth waveform can be achieved.

Figure 5(a) shows the calculated values of ${X}_{i}/{Z}_{i}$ versus modulation index. When ${m}_{i}\text{=1}\text{.142}$, the ratio of ${X}_{i}/{Z}_{i}$ is −3. For convenience, suppose the power is equally distributed on two orthogonal components (viz. $\theta \text{=}\pi /\text{4}$). Figure 5(b) gives the calculated coefficients versus bias index on condition that ${m}_{i}=1.142$ and $\theta \text{=}\pi /\text{4}$. Although the ratio of ${X}_{\text{1}}:{Y}_{2}:-{Z}_{1}$ does not fit that of Eq. (16), it can be further modified by adjusting the polarization of the incident light. If $\theta \text{=0}\text{.684}$, the expected ratio can be satisfied.

The calculated parameters and coefficients for sawtooth waveform generation are listed in Table 1. The ratio of the coefficients is very close to the ideal ratio of $\text{3}:2:\text{1}$. Figure 6 shows the simulation results of sawtooth (or reversed-sawtooth) waveform synthesis by superposing the two photocurrents.

Although the above simulation gives an approximate sawtooth waveform, the falling edge is not smooth. Furthermore, a smooth sawtooth waveform can be obtained by setting the appropriate parameters. The coefficient relation is

From Eq. (18), the following equation can be derived

With the conditions above, the two terms ($Saw1$ and $Saw2$) of Eq. (17) may present both the saw-toothed envelopes. Figure 7(a) shows the relationship between the bias index and ${X}_{i}/{Y}_{i}$ with the modulation index of 1.142. It is clear that only if the bias indices are the 1.077 and 0.492, the values of ${X}_{i}/{Y}_{i}$ can be very close to −2 and + 2 respectively. According to the results shown in Figs. 5(a) and 7(a), Eq. (19) can be achieved. Figure 7(b) presents the calculated coefficients on condition that ${m}_{i}=1.142$ and $\theta =\pi /\text{4}$.

The calculated parameters and coefficients for smooth sawtooth waveform generation are listed in Table 2. The ratios of coefficients are very close to those in Eq. (18). Thus, the sawtooth waveform can be obtained at Output 2.

Figure 8(a) shows the simulation result of the superposition of the two sawtooth waveforms expressed by Eq. (17). Figure 8(b) presents the comparison of the sawtooth waveform (dash line) and the superimposed waveform (solid line) with a normalized phase and amplitude. Although the superimposed waveform cannot be expressed by Eq. (15), it has a sawtooth-like shape and smooth edge.

It should be noted that the photocurrents ($S\text{aw}1$ and $Saw2$) in Eq. (17) are only the mathematical results, which are convenient for finding the parameters but not the actual photocurrents given by Eq. (6). Figure 9 shows the simulation results of the same waveforms generation by superimposing the two actual photocurrents.

## 3. Experiments and results

In order to verify the feasibility of the proposed scheme, the experimental demonstration is carried out. Firstly, the polarization-dependent characteristic of MZM is investigated. Here, a sinusoidal drive signal with the frequency of 3 GHz is applied, and an optical carrier suppression (OCS) modulation experiment is performed. As shown in Fig. 10(a), the spectra are detected by an optical spectrum analyzer (OSA, YOKOGAWA AQ6370C), and part of the optical carrier is left if the polarization state of the incident optical field has an angle with the principal axis. Figures 10(b) and 10(c) give the corresponding waveforms measured by an oscilloscope (Agilent 86100D Infiniium DCA-X). By comparing the two waveforms, it can be confirmed that they are both OCS signals, but the one whose polarization state disaccords with the principal axis contains a large DC component. The results indicate that the optical carrier is hardly modulated when its polarization disaccords with the principal axis. Therefore, the polarization-dependent character of MZM can be used to load signals independently on two orthogonal components of one CW source.

Next, as illustrated in Fig. 1, the generation of versatile waveforms is shown experimentally. In the experiment, we use two MZMs with the bandwidth of 10 GHz and a photodetector with the 3-dB bandwidth of 50 GHz. The CW light emitted by the LD is sent to MZM1, which is biased at the QB point and driven by a RF signal with the frequency of 3 GHz. By setting the appropriate driving voltage, a square-shaped waveform is obtained. Figure 11(a) shows the square-shaped waveform observed at Output 1. The corresponding electrical spectrum is given by Fig. 11(b), which is measured by an electrical spectrum analyzer (ESA, Agilent N9010A EXA). Here, the power ratio of the first harmonic to the third is 10.17 dBm, which is close to the theoretical value. According to the theoretical analysis, two QB point biased MZMs with the modulation index of 0.752 can be used to achieve triangular waveform generation when PC1 and PC2 are carefully adjusted to distribute the appropriate power ratio on two orthogonal components. Meanwhile, the ODL is adjusted to make the time delay of about 83 ps (corresponding to π/2 envelope phase shift of the modulated signal). Figure 11(c) shows the triangular waveform obtained at Output 2. The corresponding electrical spectrum is given by Fig. 11(d), which also agrees with the theoretical expression.

In order to confirm that the sawtooth waveform can be synthesized by two envelopes carried on two orthogonal components, the waveforms from the PBS are observed firstly as shown by Figs. 12(a) and 12(b). Figures 12(c) and 12(f) show the sawtooth (or reversed-sawtooth) waveform superimposed by these two waveforms, and the waveforms with a smaller time scale are given by Figs. 12(d) and 12(g). It is obvious that the experimental results are consistent with the simulation results. The measured falling time and rising time are 261 ps and 79 ps. It can be calculated that the ratio of the falling time to the rising time stands at about 3:1. Figures 12(e) and 12(h) show the corresponding electrical spectra.

Finally, based on the theoretical analysis, the sawtooth waveform with smooth edges can be obtained at Output 2 by adjusting the bias voltages of the MZMs and the two PCs. Figures 13(a) and 13(b) present the waveforms on the two orthogonal components. Figures 13(c), 13(d), 13(f) and 13(g) show the smooth sawtooth (or reversed-sawtooth) waveform corresponding to the simulation results shown by Fig. (9). The falling time is 232 ps and the rising time 108 ps with the corresponding ratio of about 2:1. By comparing the waveforms shown in Fig. 12 and those in Fig. 13, it can be found that the latter are smoother. The corresponding electrical spectra are given by Figs. 13(e) and 13(h).

From the experimental verification above, it can be seen that the results are consistent with the theoretical analyses and that the proposed scheme is efficient. By using one light source and two single-drive Mach-Zehnder modulators, various waveforms can be generated conveniently without using any dispersive elements or optical filters. In addition, complex spectral line manipulation is not required. Taking into account the bandwidth limitation of our measuring instrument (ESA with the maximum measurement frequency of 26.5 GHz), we only carry out the experiment of 3-GHz waveforms generation. Actually, the proposed system can offer potential for larger operation bandwidth, which is mainly subject to the bandwidth of the MZMs and PD, especially the latter, because PD must have the bandwidth larger than the highest order tone. In 2005, a PIN photodetector with the bandwidth of 120 GHz was reported [16], which means that it is possible to achieve more than 10-order harmonic tones of 10-GHz sawtooth waveform (or higher frequency triangular waveform) in our scheme.

## 4. Conclusion

A simple and efficient scheme for generating various waveforms is proposed, and the theoretical analysis and the experimental verification are carried out. Two cascaded MZMs are used to carve the CW light emitted by the LD. Due to the polarization-dependent character of MZM, the two PCs in front of the MZMs can be used to control the polarization of the incident light to achieve different spectral characteristics on the orthogonal components. In the experiment, by using sinusoidal drive signal at the frequency of 3 GHz we have successfully obtained square-shaped waveform, triangular waveform and sawtooth (or reversed-sawtooth) waveform with the repetition frequency of 3 GHz. Furthermore, an approach to generating the sawtooth waveform with smooth edges has been validated.

## Funding

National Natural Science Foundation of China (NSFC) (61465002, 61751102); High Level Innovation Talent Program of Guizhou Province, China (2015-4010); the College Innovation Talent Team of Guizhou Province, China (2014-32).

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