Generation of triangular waveforms based on a microwave photonic filter with negative coefficient

Wei Li; Wen Ting Wang; Wen Hui Sun; Wei Yu Wang; Ning Hua Zhu

doi:10.1364/OE.22.014993

1. Introduction

In recent years, there has been a growing interest in the photonic generation of triangular waveforms because of its applications in all-optical signal processing and microwave signal manipulation, e.g. optical frequency conversion, pulse compression, doubling of optical signals, and signal copying [1–3]. Many interesting approaches have been proposed to generate triangular waveforms [4–11]. For example, triangular waveforms can be generated based on frequency-to-time mapping in a dispersive element [1,4,5]. The optical spectrum of ultra-short pulses is predesigned to be a scaled version of the desired triangular waveforms. After frequency-to-time mapping, triangular waveforms are generated. All-optical triangular waveform generation has also been reported using self-convolution of an ultra-short rectangular pulse [6]. However, the limitation of using ultra-short pulses is that the generated triangular waveforms usually have small duty cycle (<1). For many application, triangular waveforms with full-duty-cycle is highly desired [3].

External modulation of a continuous wave optical signal is a promising solution to generate triangular waveforms with full-duty-cycle [7–13]. The modulated optical sidebands are manipulated in the optical domain. As a result, the microwave harmonics corresponding to the Fourier components of triangular waveforms are controllable and the desired triangular waveforms can be generated by photodetecting the tailored optical signals. Li et al. proposed a scheme using a dual-parallel Mach-Zehnder modulator (DPMZM) [7]. The two sub-MZMs of the DPMZM are driven by a fundamental microwave signal and its frequency tripling tone, respectively. In [8], the joint use of a dual-electrode (DEMZM) and a spool of dispersive fiber was proposed to generate triangular waveforms. However, the frequency of the generated triangular waveform cannot be changed for a given length of dispersive fiber. To generate a frequency tunable triangular waveform, an architecture using a MZM followed by an optical interleaver and a polarization beam combiner was proposed [9]. However, it is noted that the methods reported in [7–9] were only theoretically analyzed and simulated without providing experimental supports. Recently, triangular waveforms was experimentally generated in [10–13]. In [10], a DPMZM and a 90° hybrid electrical coupler were used to generate triangular waveforms. We have reported the generation of triangular waveforms using a DPMZM and an optical bandpass filter [11]. Sophisticated bias control is required since there are three biases to be controlled in a DPMZM. Liu et al. reported a method to generate triangular waveforms using a polarization modulator incorporated in a Sagnac loop [12]. However, the system might suffer from stability problem due to the use of Sagnac interferometer and complicated polarization controls.

In this paper, we propose and experimentally demonstrate a new method to generating full-duty-cycle triangular waveforms based on a microwave photonic filter (MPF) with negative coefficient. It is known that the Fourier series expansion of a triangular waveform has only odd-order harmonics. In this work, the undesired even-order harmonics are suppressed by the MPF that has a periodic transmission response, while the desired odd-order harmonics are well kept. The triangular waveforms are generated in two ways. One is that the conventional MZM is biased at quadrature point. The other one is that the MZM is biased at maximum or minimum transmission point. Triangular waveforms at fundamental frequency can be generated by biasing the MZM at the quadrature point. However, we find that a broadband 90° microwave phase shifter has to be used after photodetection to adjust the phases of odd-order harmonics. In the other case, frequency doubling triangular waveforms can be generated by biasing the MZM at the maximum or minimum transmission point. This method is more promising since the broadband 90° microwave phase shifter is no longer required but it is more power consuming because the required power of the input sinusoidal microwave signal is more than that in the previous case. The proposed approach is theoretically analyzed and experimentally verified. Triangular waveforms at 5 GHz using a 5 GHz or 2.5 GHz sinusoidal microwave signal have been experimentally generated. The frequency tunable range of the generated triangular waveforms are mainly limited by the bandwidth of the devices used in our experiment.

2. Principle

The schematic diagram of the proposed triangular waveform generator is shown in Fig. 1(a). An optical carrier from a laser diode (LD) is modulated by a MZM. A sinusoidal microwave signal is applied to the MZM from a vector network analyzer (VNA). The intensity-modulated signal is divided into two branches by an optical coupler and sent to the two input ports of a balanced photodetector (BPD), respectively. A tunable optical delay line (TODL) is inserted in one of the two branches. The architecture shown in Fig. 1(a) is equivalent to a MPF with negative coefficient [14]. It is known that the Fourier series expansion of a triangular waveform T(t) can be expressed as

T (t) = D C + \sum_{m = 1, 3, 5}^{\infty} \frac{1}{m^{2}} \cos (m Ω t)

where Ω is the fundamental angular frequency. It can be seen that a triangular waveform consists of only odd-order harmonics. Actually, Eq. (1) can be rewritten as

T (t + t_{0}) = D C + \sum_{m = 1, 3, 5}^{\infty} \frac{1}{m^{2}} \cos (m Ω t + m Ω t_{0})

where t₀ denotes constant time. Equation (2) shows that the shape of a triangular waveform is unchanged if phase shift mΩt₀ (m = 1,3,5…) is introduced to the harmonic of the triangular waveform. The key points to generate triangular waveforms are summarized as: first, the electrical signal has only odd-order harmonics. Second, the amplitudes of the odd-order harmonics satisfy 1/m² (m = 1,3,5…). Finally, the phases of the harmonics have to be mΩt₀ (m = 1,3,5…).

Fig. 1 Schematic diagrams of (a) the proposed triangular waveform generator, (b) and (c) the principle. LD: laser diode, MZM: Mach-Zehnder modulator, TODL: tunable optical delay line, BPD: balanced photodetector, VNA: vector network analyzer, ESA: electrical spectrum analyzer, OSC: oscilloscope.

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In our approach, the undesired even-order harmonics are removed by a MPF with periodic transmission response. Figures 1(b) and 1(c) show the schematic diagrams of the operational principle using a MPF with positive and negative coefficient, respectively. As can be seen from Fig. 1(b), although the second-order harmonic is removed by the notch of the MPF with positive coefficient, the fourth-order harmonic still exists. Moreover, the desired odd-order harmonics suffer from attenuation since they are not located at the peaks of the transmission response. On the other hand, the MPF with negative coefficient is preferred as shown in Fig. 1(c). The undesired even-order harmonics are suppressed since they fall into the notches of the MPF, while the desired odd-order sideband are well kept because they are located at the peaks of the MPF. The amplitudes of the harmonics can be controlled by the power of the sinusoidal microwave signal applied to the MZM. The phase requirement for generation of a triangular waveform will be discussed in detail in the following parts. We will discuss about two cases. One is that the MZM is biased at quadrature point. The other one is that the MZM is biased at maximum or minimum transmission point.

Mathematically, the optical field at the output of the MZM is given by

E (t) = \frac{1}{2} \exp (j ω_{0} t) {\exp [j β \cos (ω_{m} t) + j \frac{φ}{2}] + \exp [- j β \cos (ω_{m} t) - j \frac{φ}{2}]}

where ω₀ and ω_m are the angular frequencies of the optical carrier and the sinusoidal microwave signal, respectively. β = πV_m/V_π is the modulation index of the MZM, V_m and V_π are the amplitude of the microwave signal and the half-wave voltage of the MZM, respectively. φ = πV_bias/V_π is the phase shift between the two arms of the MZM, which is controlled by the bias V_bias of the MZM. Applying Jacobi Anger expansion to Eq. (3), we have

E (t) = \frac{1}{2} \exp (j ω_{0} t) {\sum_{n = - \infty}^{\infty} j^{n} J_{n} \exp (j n ω_{m} t + j \frac{φ}{2})] + \sum_{n = - \infty}^{\infty} {(- 1)}^{n} j^{n} J_{n} \exp (j n ω_{m} t - j \frac{φ}{2})}

where J_n = J_n(β) is the Bessel function of the first kind of order n. The optical signal is divided into two branches by a 3-dB optical coupler and sent to the two input ports of the BPD, respectively. The photocurrent at the output of the BPD is given by

\begin{matrix} i (t) = i_{p d 1} (t) - i_{p d 2} (t) \\ \propto \frac{1}{2} E (t) \cdot E^{*} (t) - \frac{1}{2} E (t + T_{0}) \cdot E^{*} (t + T_{0}) \end{matrix}

where i_pd1(t) and i_pd2(t) are the photocurrents from the two PDs of the BPD, respectively. T₀ is the time difference between the two optical branches which is controlled by the TODL.

First, we consider the case that the MZM is biased at quadrature point, i.e. φ = π/2. It was reported that two Fourier components can make a good approximation of a triangular wave-form since the powers of the higher order components decrease fast [7–12]. Thus, we consider the generated microwave harmonics as well as the optical sidebands up to the third-order ones. Equation (5) can be rewritten as

\begin{matrix} i (t) \propto D C + \frac{a_{1}}{2} [\cos (ω_{m} t) - \cos (ω_{m} t + ω_{m} T_{0})] \\ + \frac{b_{1}}{2} [\cos (2 ω_{m} t + π) - \cos (2 ω_{m} t + π + 2 ω_{m} T_{0})] \\ + \frac{c_{1}}{2} [\cos (3 ω_{m} t + π) - \cos (3 ω_{m} t + π + 3 ω_{m} T_{0})] \end{matrix}

where a₁ = 2J₀J₁–2J₁J₂ + 2J₂J₃, b₁ = 2J₀J₂–J₁J₁ + 2J₁J₃, and c₁ = 2J₀J₃ + 2J₁J₂. Comparing Eq. (6) with Eq. (2), the dominant second-order harmonic at 2ω_m has to be suppressed. Thus, we have 2ω_mT₀ = (2k + 1)π (k is an integer). Moreover, c₁/a₁ = 1/9 has to be satisfied. Equation (6) can be simplified as

i (t) \propto D C + a_{1} \cos (ω_{m} t) + a_{1} / 9 \cos (3 ω_{m} t + π)

As can be seen from Eq. (7), the second-order harmonic is eliminated by the MPF. However, the phases for the fundamental tone and the third-order harmonic are 0 and π, respectively, which violate the requirement for generation of a triangular waveform (see Eq. (2)). If a broadband 90° microwave phase shifter is attached after the BPD to introduce a 90° phase shift to both the fundamental tone and the third-order harmonic, Eq. (7) is rewritten as

i (t) \propto D C + a_{1} \cos (ω_{m} t + π / 2) + a_{1} / 9 \cos (3 ω_{m} t + 3 π / 2)

It is apparent that Eq. (8) fully meets the requirement for generation of a triangular waveform as shown in Eq. (2). Thus, a triangular waveform at fundamental frequency Ω = ω_m can be generated.

In the second case, we consider that the MZM is biased at the maximum (or minimum) transmission point, i.e. φ = 0 (or π). The odd-order (or even-order) optical sidebands are suppressed. If this optical signal is directly detected by a PD, the frequency components at Ω = 2ω_m, 2Ω = 4ω_m, 3Ω = 6ω_m, 4Ω = 8ω_m… are produced. In order to generate a triangular waveform, the harmonics at 2Ω = 4ω_m, 4Ω = 8ω_m… have to be eliminated using the MPF. In this case, we consider microwave harmonics and the optical sidebands up to the sixth-order ones to include the frequency component at 3Ω = 6ω_m. Equation (5) is rewritten as

\begin{matrix} i (t) \propto D C + \frac{a_{2}}{2} [\cos (2 ω_{m} t + π) - \cos (2 ω_{m} t + π + 2 ω_{m} T_{0})] \\ + \frac{b_{2}}{2} [\cos (4 ω_{m} t) - \cos (4 ω_{m} t + 4 ω_{m} T_{0})] \\ + \frac{c_{2}}{2} [\cos (6 ω_{m} t + π) - \cos (6 ω_{m} t + π + 6 ω_{m} T_{0})] \end{matrix}

where a₂ = 2J₀J₂ + 2J₂J₄ + 2J₄J₆, b₂ = 2J₀J₄ + J₂J₂ + 2J₂J₆, and c₂ = 2J₀J₆ + 2J₂J₄. Comparing Eq. (9) with Eq. (2), 4ω_mT₀ = (2k + 1)π (k is an integer) and c₂/a₂ = 1/9 have to be satisfied. So, Eq. (9) can be simplified as

i (t) \propto D C + a_{2} \cos (2 ω_{m} t + π) + a_{2} / 9 \cos (6 ω_{m} t + 3 π)

As can be seen from Eq. (10), it is fully compliant with the requirement for generation of a triangular waveform. It is worth noting that it does not need a broadband 90° microwave phase shifter as the quadrature-biased case. Moreover, a frequency doubling triangular waveform is generated at Ω = 2ω_m.

For the above two cases, the amplitude ratio between the third-order harmonic and the fundamental tone, i.e. c₁/a₁ and c₂/a₂, should be 1/9 to satisfy the requirement of generating a triangular waveform. Figure 2 shows this amplitude ratio versus the modulation index of the MZM, β. As can be seen, the modulation index of the MZM should be 0.753 and 1.948 rad for the quadrature-biased MZM and the maximum-biased MZM, respectively. It is found that the maximum-biased MZM is more power consuming since the required microwave power is larger than the quadrature-biased MZM.

Fig. 2 Amplitude ratio between the third-order harmonic and the fundamental tone versus the modulation index of the MZM, β.

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3. Experiment

We also carried out experiments based on Fig. 1(a) to verify the feasibility of the proposed approach. An optical carrier at 1550 nm from a LD was fiber-coupled to a MZM which has a bandwidth of 40 GHz and a V_π of ~3.5 V. After splitting by a 3-dB optical coupler, the two intensity-modulated optical signals are sent to the two input ports of a BPD, respectively. The 3-dB bandwidth of the BPD is ~20 GHz. The sinusoidal microwave signal was provided by a VNA which has a bandwidth of 40 GHz. The electrical spectrum was measured using a 40 GHz electrical spectrum analyzer (ESA). The waveform was captured by an oscilloscope (OSC).

First, the MZM was biased at the quadrature point. We tried to generate a triangular waveform at 5 GHz. Thus, the free spectrum range (FSR) of the MPF was tuned to be 10 GHz by adjusting the TODL. The measured transmission response of the MPF is shown in Fig. 3.. It has a periodic bandpass transmission response. The peaks are located at 5, 15, and 25 GHz, while the notches are located at 10, 20, and 30 GHz. We applied a 5 GHz sinusoidal microwave signal to the MZM. The power of the microwave signal was set to be ~12 dBm. The uneven frequency response of the BPD was also considered when we chose the microwave power. The measured electrical spectrum is shown in Fig. 4(a).The even-order harmonics at 10 and 20 GHz are significantly suppressed by the MPF, while the odd-order harmonics are well kept. The third-order harmonic at 15 GHz is 19.15 dB lower than the fundamental tone, which is very close to the ideal value of 19.08 dB (corresponding to the amplitude ratio of 1/9). Thanks to the MPF, the dominant second-order harmonics at 10 GHz is suppressed to be 31 dB lower than the fundamental tone. Thus, the amplitudes of the generated harmonics meet the requirement of generating a triangular waveform.

Fig. 3 Measured transmission response of the MPF with a FSR of 10 GHz.

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Fig. 4 Measured (a), (c) electrical spectra and (b), (d) the waveforms when the MZM is biased at the quadrature point and a 90° broadband microwave phase shift was attached after the BPD (see Figs. 4(c) and 4(d)) or not (see Figs. 4(a) and 4(b)).

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The measured (solid line) and simulated (dashed line) waveforms are shown in Fig. 4(b). As discussed previously, the phases of the fundamental tone and the third-order harmonic do not meet the requirement of generating a triangular waveform, though the amplitudes are fully compliant with the requirement. As shown in Fig. 4(b), the generated waveform is not a triangular-shaped pulse. The measured result agrees well with the simulated one based on Eq. (7). As analyzed in the theoretical part, a 90° phase shift has to be added to both the fundamental tone and the third-order harmonic to generate a triangular waveform. In our experiment, a 90° hybrid 3-dB electrical coupler which has a bandwidth from 1 to 17 GHz was attached at the output of the BPD. The measured electrical spectrum is shown in Fig. 4(c). As can be seen, the frequency components within the frequency range from 1 to 17 GHz is attenuated by 3 dB due to the use of the 90° hybrid 3-dB electrical coupler, while the frequency components out of this bandwidth are significantly suppressed. The second-order and the third-order harmonics are 31 and 19.5 dB lower than the fundamental tone, respectively. The measured (solid line) waveform is shown in Fig. 4(d). A triangular waveform at 5 GHz was successfully generated. The measured triangular waveform fits well with the simulated one (dashed line). The root-mean-square error (RMSE) between the measured and the simulated result is 4.3e-5. The RMSE between the measured and the ideal triangular waveform is 8.8e-4. Thus, two Fourier components can make a good approximation of a triangular waveform.

Next, the MZM was biased at the maximum transmission point. As discussed previously, it could generate a frequency doubling triangular waveform. Thus, to generate a triangular waveform at 5 GHz, the sinusoidal microwave signal applied to the MZM was set to be 2.5 GHz with power of ~20 dBm. The power consumption is ~8 dB higher than the quadrature-biased case. In order to show the effect of the MPF on the generation of triangular waveform. The MPF was first removed and the optical signal at the output of the MZM was directly sent to a PD with a bandwidth of 40 GHz. The measured electrical spectrum is shown in Fig. 5(a). Since the MZM was biased at the maximum transmission point, the frequency components at 5, 10, 15, and 20 GHz were generated using the 2.5 GHz sinusoidal microwave signal, while other frequency components were suppressed. However, due to the lack of MPF, the second-order harmonics at 2Ω = 10 GHz is only 6.4 dB lower than the frequency components at Ω = 5 GHz and even 12.6 dB higher than the harmonics at 3Ω = 15 GHz. The measured (solid line) and simulated (dashed line) waveforms are shown in Fig. 5(b). A full-duty-cycle triangular waveform was not generated due to the dominant frequency at 2Ω = 10 GHz. Then, the MPF was rebuilt. The TODL was kept unchanged to obtain a MPF with FSR of 10 GHz as the previous case. As discussed in Section 2, no broadband microwave phase shifter is required in this case. Thus, the electrical signal at the output of the BPD was directly measured. The measured electrical spectrum is shown in Fig. 5(c). As can be seen, the undesired frequency components at 2Ω = 10 GHz was suppressed to be 30 dB lower than the component at Ω = 5 GHz. The frequency component at 3Ω = 15 GHz is 19.09 dB (close to the ideal value of 19.08 dB) lower than that at Ω = 5 GHz. The measured (solid line) waveform is shown in Fig. 5(d). A frequency doubling triangular waveform at Ω = 5 GHz with full-duty-cycle was successfully generated. The simulated waveform (dashed line) shown in Fig. 5(d) agrees well with the measured one. The RMSE between the measured and the simulated result is 7.4e-4, while the RMSE between the measured and the ideal triangular waveform is 2.2e-3.

Fig. 5 Measured (a), (c) electrical spectra and (b), (d) the corresponding waveforms when the MZM was biased at the maximum transmission point and the MPF was used to suppress the even-order harmonics (see Figs. 5(c) and 5(d)) or not (see Figs. 5(a) and 5(b)).

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From the above experiments, it is found that the approach using a maximum- (or minimum-) biased MZM is more promising compared to the quadrature-biased one, though it is more power consuming. Not only a broadband microwave phase shifter is no longer required, but also a frequency doubling triangular waveform can be generated. Finally, to show the frequency tunability of the proposed scheme, we tried to generate triangular waveform at other frequency using the more promising approach (maximum-biased MZM). The frequency of the sinusoidal microwave signal was changed to 3 GHz. Thus, a frequency doubling triangular waveform at 6 GHz could be expected. The FSR of the MPF was changed to be 12 GHz by adjusting the TODL. The measured transmission response of the MPF is shown in Fig. 6(a).The notches of the MPF are located at 12 and 24 GHz, while the peaks are located at 6 and 18 GHz. The measured electrical spectrum is shown in Fig. 6(b). The undesired harmonics are suppressed to be 27 dB lower than the frequency component at Ω = 6 GHz, while the frequency component at 3Ω = 18 GHz is 18.54 dB lower than that at Ω = 6 GHz. A frequency doubling triangular waveform at Ω = 6 GHz was successfully generated as shown in Fig. 6(c). The measured (solid line) and the simulated (dashed line) results agree well with each other. The RMSE between the measured and the simulated result is 5.3e-4. The RMSE between the measured and the ideal triangular waveform is 2.8e-3.

Fig. 6 Measured (a) transmission response of the MPF with a FSR of 12 GHz, (b) electrical spectrum, and (c) triangular waveform at 6 GHz.

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As can be seen from Figs. 5(d) and 6(c), the generated triangular waveforms have a little distortion. Moreover, the RMSE for the MZM maximum-biased case is larger than that of the quadrature-biased case. This can be attributed to the limited extinction ratio of the MZM (~20 dB). The odd-order optical sidebands cannot be fully suppressed when the MZM is biased at the maximum transmission point. As a result, the undesired microwave harmonics were generated after photodetection. This point has been supported by the experimental results as shown in Figs. 5(a), 5(c), and 6(b). Taking Fig. 6(b) as an example, the undesired harmonics at 9 and 15 GHz are 27 dB lower than the desired one at 6 GHz. For an ideal MZM with infinite extinction ratio, the undesired harmonics at 3, 9, and 15 GHz do not exist as predicted by Eq. (9). According to the theory of MPF, the mainlobe-to-sidelobe ratio increases with the tap number [14]. Therefore, this problem can be overcome by using a multi-tap MPF with negative coefficient. It is worth noting that the frequency tunable range of the proposed triangular waveform generator is mainly limited by the bandwidth of the devices used in our experiment.

4. Conclusion

We have theoretically and experimentally demonstrated a novel approach to generating a full-duty-cycle triangular waveform based on a MPF with negative coefficient. The undesired harmonics are removed by the MPF with periodic transmission response, while the desired ones are well kept. A triangular waveform at fundamental frequency is generated by setting the bias of the MZM at quadrature point. However, a broadband 90° microwave phase shifter has to be used after the BPD to adjust the phases of odd-order harmonics. On the other hand, a frequency doubling triangular waveform can be generated if the MZM is biased at the maximum or minimum transmission point. In this case, the broadband microwave phase shifter is no longer needed but it is more power consuming. We have experimentally generated triangular waveform at 5 GHz using a sinusoidal microwave signal at 5 or 2.5 GHz, respectively. Moreover, the frequency tunability of the proposed scheme have also been verified. A triangular waveform at 6 GHz has been successfully generated using a 3-GHz microwave signal. The frequency tunable range of the triangular waveform is mainly restricted by the bandwidth of the devices used in our experiment. In addition, the performance of the triangular waveform generator can be further improved by using a multi-tap MPF with negative coefficient.

It is worth noting that the proposed method is designed to generate waveforms consisting of only odd-order harmonics, e.g. triangular waveforms or square waveforms [11,12]. Unfortunately, it is not possible to generate waveforms which have both odd-order and even-order harmonics, e.g. sawtooth even arbitrary waveforms.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under 61377069, 61335005, 61108002, 61321063, and 61090391.

References and links

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Generation of triangular waveforms based on a microwave photonic filter with negative coefficient

Abstract

1. Introduction

2. Principle

3. Experiment

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (6)

Equations (10)

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