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Abstract
A photonic method of sawtooth waveform generation by using one single-drive Mach-Zehnder modulator is proposed and experimentally demonstrated. Depending on the polarization-sensitive characteristic of the modulator, the modulation sidebands and optical carrier can independently exist on two orthogonal polarization directions. Therefore, the required Fourier components can be manipulated on two polarization dimensions separately, and the superposition of the orthogonal optical envelopes synthesize a sawtooth waveform in time domain. The feasibility of the scheme is theoretically analyzed. In the experiment, sawtooth waveforms with full duty cycle at 3, 5, and 8 GHz are obtained, which agree with the simulation results well.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Generation of microwave arbitrary waveforms is an important research field due to their applications in radar, sensors, wireless communication, and signal processing [1–4]. Conventionally, microwave waveforms are generated by electrical methods, but the limited bandwidth confines signal generation with high frequency. Fortunately, microwave photonic techniques exhibit advantages of large bandwidth, immunity to electromagnetic interference and so on, which can be good candidates for microwave arbitrary waveform generation.
In recent decades, based on Fourier synthesis method or time-domain synthesis method, many photonic solutions for microwave arbitrary waveform generation have been proposed and successfully demonstrated [5–8]. Fourier synthesis is a popular method, in which the spectrum lines of the optical comb are spatially separated by a spectral disperser and the amplitude and phase of each spectrum line are controlled by a spatial light modulator. After a spectrum combiner, the desired arbitrary waveform is generated [5,6]. However, the typical Fourier synthesis causes a complex system and it is sensitive to environment fluctuation. By carving and superimposing optical field envelopes in time domain, time-domain synthesis method presents specific advantages for generating certain waveforms, in which the expected signals can be achieved without manipulating spectrum lines [7,8].
In practice, external modulation technique is widely employed in both Fourier synthesis and time-domain synthesis, because the modulation process can produce controllable Fourier components in frequency domain or it can be regarded as a graver for signal envelope in time domain. There are many works of microwave arbitrary waveform generation with external modulation technique have been reported, but some limitations are still found [9–11]. In the cases of Fourier synthesis, external modulation provides limited harmonics, which leads to most of the previous works focused on triangular waveform generation due to only the second-order approximation required. In other words, because sawtooth waveform generation need manipulate more Fourier components, it is difficult to be obtained by a simple way. For example, sawtooth waveform generation by using a polarization modulator inserted Sagnac loop was reported in [12]. However, this scheme requires high drive voltage, and an optical filter has to be employed to remove unwanted sidebands. Sawtooth waveforms generation by utilizing a Dual-Drive Mach-Zehnder modulator (MZM) or a dual-parallel Mach-Zehnder modulator have been proposed in [13,14]. But the special devices and complicated bias control make the schemes inconvenient. Sawtooth waveform generation by time-domain synthesis shows some advantages. Jiang et al. demonstrated two schemes of sawtooth waveform generation by carving and overlapping optical field envelopes via two cascade single-drive MZMs [7,15]. The quality of generated signal is good, but two modulation steps increase the cost and complexity.
Although external modulation technique is applied on sawtooth waveform generation, only the low-order approximation can be achieved. In [12–15], third-order approximate sawtooth waveforms are obtained because the correct phase and weight of the fourth-order component can not be reached. Consequently, the generated sawtooth waveforms always exhibit undulating edges, which is harmful to most of applications.
In this paper, we propose a simple scheme of tunable sawtooth waveform generation based on only one single-drive MZM, in which the advantages of Fourier synthesis and time-domain synthesis are effectively taken. According to the polarization sensitivity in a LiNbO3 MZM [16], the modulation sidebands and optical carrier can be independently controlled on two orthogonal polarization direction, if the polarization state of incident light has an angle with the principal axis of the modulator. Therefore, by power splitting and harmonics manipulating on two polarization branches, sawtooth waveform with full duty cycle can be obtained by the superposition of two orthogonal optical envelopes. One fact is that the generated sawtooth waveform has a tolerance range under the low-order approximation, which allows us to optimize the linearity of the edge by properly changing the power ratio among the harmonics [17]. To balance the smoothness and accuracy of the waveform, theoretical analysis and simulations are performed. In the experiment, sawtooth waveforms with repetition frequency of 3, 5, and 8 GHz are obtained, which agree with the theoretical expection.
2. Operation principle
As we know, the Fourier expansion of sawtooth waveform can be expressed as
Previous work has confirmed that the first three components can contribute an acceptable waveform [15], which means Eq. (1) can be simplified asFigure 2 is the schematic diagram of the proposed sawtooth waveform generator. From an external cavity laser (ECL), a continuous wave (CW) can be expressed as
where ${E_0}$ is the optical field amplitude, and ${\omega _0}$ is the angular frequency of the optical field. After adjusting the polarization state by PC1, this CW is modulated by a sinusoidal signal ${V_m}\cos {\omega _m}t$ via a MZM, where ${V_m}$ and ${\omega _m}$ are the amplitude and frequency of the drive signal. For convenience, the principal axis and the orthogonal axis of the MZM are denoted by X-axis and Y-axis. When the MZM is biased at minimum transmission point (MITP) with a voltage of ${V_{bias}}$, and the polarization state of the CW has an angle $\alpha $ relative to the X-axis, the light field at point 2 can be written as
Fig. 2. The configuration and optical spectra evolution of the proposed scheme. ECL: external cavity laser, PC: polarization controller, MZM: single-drive LiNbO3 Mach-Zehnder modulator, EDFA: erbium-doped fiber amplifier, OC: optical coupler, POL: polarizer, ODL: optical delay line, PBC: polarization beam combiner.
After a 3-dB optical coupler (OC), the power of the modulated optical field is split into upper and lower branches. In the upper branch, the modulation sidebands on X direction are rotated to parallel with the polarizer (POL1). Thus, the optical carrier is filtered out and the transmitted sidebands produce even-order harmonics of the drive signal in a photodetector (PD). By applying Jocabi Anger expansion, the optical field at point 3 can be given by
In the lower branch, an angle $\theta$ between the polarization state of the optical carrier and POL2 is set by aligning PC3, which means part of optical power on the modulation sidebands and optical carrier can be projected on the same polarization direction. In this case, the required odd-order harmonics in Eq. (2) can be obtained at point 4. The optical field and corresponding photocurrent are written as
Under the low-order approximation condition, the waveform strictly matching the Fourier expansion might not be the best choice, as shown in Fig. 1. Fortunately, the waveforms have a tolerance range, which means the shape of the waveform can be optimized by changing the weight of the components [17]. Of cause, the fitting degree will be also changed. To balance the smoothness and accuracy of the waveform, some allowable parameters and the corresponding R-squared are listed in Table 1, and the simulation results are illustrated by Fig. 4. It needs to be pointed out that the fourth-order component is considered as an interference term in the result. A summary from even more simulations, when the ratio between the first-order and second-order components is larger than 2.5, and the smaller weights of the third-order and the fourth-order components are taken, the smoother edges of the waveforms can be approached, but the R-squared is declining in general. Taking both of the smoothness and fitting degree into account, the coefficients within the blue region around point A in Fig. 4 are the better choices.
Table 1. The R-squared with different sets of coefficients.
Suppose the status of $\beta \textrm{ = }1.386$, $\alpha \textrm{ = }72^\circ$, $\theta \textrm{ = 78}^\circ$, $\varepsilon \textrm{ = }0.672$, $\delta \textrm{ = }0.84$ and ${{\phi \textrm{ = }\pi } \mathord{\left/ {\vphantom {{\phi \textrm{ = }\pi } 4}} \right.} 4}$ are set, the situation at point A in Fig. 4 can be reached and the corresponding photocurrent is
The simulation result is shown by Fig. 5(a). When $\phi $ is changed to be ${{3\pi } \mathord{\left/ {\vphantom {{3\pi } 4}} \right.} 4}$ and other parameters are held, the reversed-sawtooth waveform is obtained, as shown by Fig. 5(b). Obviously, compared with the case of ideal third-order approximation, the generated waveforms have much better profile with only 4.47% fitting degree degradation.
3. Experiments and results
The experimental verification based on Fig. 2 is carried out. Firstly, we investigate the polarization sensitivity of MZM. A CW is emitted by an ECL and sent into a single-drive MZM biased at MITP. The drive signal is a sinusoidal signal with frequency of 5 GHz, and the angle $\alpha $ between the polarization state of CW and the principal axis of MZM is controlled by PC1. From the optical spectrum analyzer (OSA, YOKOGAWA AQ6370C) and oscilloscope (Agilent 86100D Infiniium DCA-X), the optical spectra and waveforms of modulated signals can be observed. In the case of $\alpha \textrm{ = }0$, the optical carrier is well suppressed and a frequency-doubled sinusoidal signal is obtained, which are shown by Fig. 6(a) and Fig. 6(c), respectively. By aligning PC1 to set $\alpha \ne 0$, the optical carrier is emerged again, as shown in Fig. 6(b). However, it can not beat with the modulation sidebands due to the orthogonal polarization, and only contributes a DC component on the waveform in Fig. 6(d). In the following experiment, the angle $\alpha $ is adjusted to be around $72^\circ$.
Fig. 6. Measured optical spectra and waveforms biased at MITP. (a), (c) The spectrum and waveform with $\alpha \textrm{ = }0$. (b), (d) The spectrum and waveform with $\alpha \ne 0$.
After an EDFA, the amplified optical field is divided into two parts by a 3-dB OC. In the upper branch, the polarization state of the modulation sidebands is controlled by PC2 to coincide with the POL1. Therefore, the optical carrier is excluded and the beat signal gives the second and the fourth items of the Fourier expansion. The corresponding optical spectrum, electrical spectra and waveform are measured at point 3, as shown by Figs. 7(a)–7(c). In the lower branch, the polarization of optical carrier and sidebands are rotated and part of them are projected into the same polarization direction as POL2. The measured optical spectrum, electrical spectra and waveform are given by Figs. 7(d)–7(f). Here, the power ratio between optical carrier and modulation sidebands can be adjusted by PC3, and the beat signal contributes the first-order and the third-order harmonics with power ratio of 20.18 dB, which approaches the requirement of Fourier expansion. In addition, the power of the second-order harmonic is 30.79 dB lower than that of the first-order component, which supports the approximation condition in Eq. (10).
Fig. 7. Experimental results observed at point 3 and point 4. (a-c) The optical spectrum, electrical spectrum and waveform at point 3. (d-f) The optical spectrum, electrical spectrum and waveform at point 4.
To synthesize sawtooth waveform in time domain without optical interference, a PBC is employed to implement the envelope superposition, and a phase shift of ${\pi \mathord{\left/ {\vphantom {\pi 4}} \right.} 4}$ is introduced by the ODL. Meanwhile, PC4 and PC5 are used to control the transmission power for each branch. In Fig. 8(a), the generated sawtooth waveform is illustrated, and the corresponding electrical spectra is shown by Fig. 8(c). One can find from the electrical spectrum that the first four harmonics have power ratio approaching the calculate value. Similarly, when a phase shift of ${{3\pi } \mathord{\left/ {\vphantom {{3\pi } 4}} \right.} 4}$ between two branches is set, reversed-sawtooth waveforms can be obtained, as shown by Fig. 8(b). The corresponding electrical spectra is shown by Fig. 8(d). Clearly, the experimental result agree with the theoretical predication in Eq. (13) well.
Fig. 8. Measured waveforms and electrical spectra. (a), (b) The optimized sawtooth and reversed sawtooth waveforms with repetition frequency of 5 GHz, and (c), (d) the corresponding electrical spectra of (a) and (b).
Finally, for verifying the tunability of the system, the frequencies of drive signal are changed to be 3 and 8 GHz respectively. Consequently, sawtooth waveforms at 3 GHz and 8 GHz are achieved, as shown by Figs. 9(a)–9(d).
Fig. 9. Measured waveforms. (a) 3-GHz sawtooth waveform. (b) 3-GHz reversed-sawtooth waveform. (c) 8-GHz sawtooth waveform. (d) 8-GHz reversed-sawtooth waveform.
It can be seen from the above verification that the experimental results are consistent with the theoretical analysis. In the experiment, we only demonstrate sawtooth waveform with the highest frequency of 8 GHz, because the maximum measurement bandwidth of ESA is 26.5 GHz. Nevertheless, the proposed scheme can achieve the waveforms with higher frequency, where the bandwidths of MZM and PD are the primary limitation.
4. Summary
In conclusion, a method of tunable sawtooth waveform generation based on only one single-drive MZM is proposed. Theoretically, the operation principle is analyzed, and the concept of fitting degree is introduced to optimize and evaluate the generated siganl. Experimentally, by utilizing polarization sensitivity of MZM, the Fourier components can be independently manipulated on two orthogonal polarization dimensions. The sawtooth waveform can be finally generated through the superposition of these two preparatory optical envelopes. Sawtooth pulses with frequency of 5 GHz are obtained in experiment, which is in accord with the simulation result. By simply changing the frequencies of drive signals, the tunability is demonstrated. Since the system contains only one light source, one single-drive MZM and no any optical filter, it exhibits great benefits on cost and convenience.
Funding
National Natural Science Foundation of China (61835003); Startup Project for High-level Talents of Guizhou Institute of Technology (2015-4010); Guizhou University (2018-5781-1); Guizhou Science and Technology Department (2016-2324).
Disclosures
The authors declare no conflicts of interest.
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