## Abstract

A theoretical model and its calculation method are proposed to simulate an actively mode-locked optoelectronic oscillator (OEO) based on electric amplitude modulation. The model includes electric amplitude modulation to achieve mode locking and convolution of electric signal and filter impulse response function to achieve mode selection. Numerical simulation is carried out through enhancing the calculating time window to an integral multiple of the roundtrip time and employing pulse tracking method with a precise delay. Through using this model, the waveform, the spectrum and the phase noise characteristic of the generated microwave pulse train from an actively mode-locked OEO are numerically simulated, where the simulation results fit in with the experimental results. This model can be used to design an actively mode-locked OEO based on electric amplitude modulation. More importantly, it is favorable for studying the dynamic process in an actively mode-locked OEO, which is difficult to grasp by carrying out an experiment.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Microwave oscillators are indispensable components in communication, radar and electronic measurement systems [1–3]. Mainstream electronic oscillators are suffering from bad phase-noise performance in the high frequency range owing to the well-known electronic bottleneck, which sets an insurmountable limit to the microwave system performance [4]. As an alternative, an optoelectronic oscillator (OEO) is recognized as a promising candidate to generate microwave signals with an ultra-low phase noise in a broad frequency range [5–7]. In order to meet the requirement of low-jitter microwave pulse train generation in applications such as pulse Doppler radars, time-domain mode-locking technique has been introduced into OEO recently [8–10]. In a mode-locked OEO, the longitudinal modes of the ring cavity in the gain spectrum are locked with a constant phase, which can be coherently superimposed in the time domain to form a microwave pulse train with a low time jitter. In our previous work [10], an actively mode-locked OEO was experimentally demonstrated by adding an electric amplitude modulator into the cavity to provide periodic loss modulation, where the externally-applied electric signal frequency is set to be equal to an integral multiple of the free spectral range (FSR) of the cavity. The most fascinating advantage of this scheme is that it can greatly reduce the close-to-carrier phase noise (e.g., the phase noise at 100-Hz frequency offset is 30-dB lower than that in a conventional OEO in [10]), which is favorable for improving the velocity detection sensitivity of a pulse Doppler radar. Nevertheless, the mechanism behind the mode locking process has not been clearly revealed. In addition, the performance of an actively mode-locked OEO is sensitive to the cavity parameters such as the net gain spectrum, the externally-applied electric signal frequency and the FSR of the cavity, which is difficult to summarize through carrying out an experiment. Therefore, it is of vital importance to establish a theoretical model to find out the inner mechanism of an actively mode-locked OEO, and to predict its behaviors under various cavity parameters and oscillating states.

In past years, several theoretical models have been established with regard to conventional OEOs [5,11–13]. In [5], Yao and Maleki employ a sustainable quasi-linear theory to model a single-tone OEO, where the calculation results agree with the experimental results in the linear region. Chembo et al. investigate the nonlinear dynamics in OEOs based on a delay-differential equation [11]. The proposed model is feasible even if the modulator works in the highly nonlinear region. Nevertheless, the bandpass filter in the OEO cavity is simply approximated to be a 2^{nd}-order system, which may introduce waveform distortion if it is used to model an actively mode-locked OEO. Levy et al. established a model to analyze the temporal envelope evolution of the microwave signal in an OEO, where the calculation is carried out in the time domain [12]. The main problem of this method lies in that the rough truncation of the temporal impulse response of the bandpass filter results in a poor filtering property, which may introduce a large time jitter to the output signal if it is employed to model an actively mode-locked OEO. Levy et al. also carried out a theoretical study on a passively mode-locked OEO based on a saturable electric amplifier, where the simulation results fit in with the experimental results [13]. However, this model cannot be directly used to characterize an actively mode-locked OEO.

In this paper, a theoretical model and its numerical calculation method are proposed to simulate an actively mode-locked OEO. The model includes electric amplitude modulation to achieve mode locking, and convolution of electric signal and filter impulse response function to achieve mode selection. Simulation is carried out through expanding the calculating time window to an integral multiple of the roundtrip time, and employing pulse tracking method with a precise delay. Through using this model, the waveform, the spectrum and the phase noise characteristic of the generated microwave pulse train from an actively mode-locked OEO are obtained, which fit in with the experimental results.

## 2. Theoretical model

Figure 1 shows the schematic diagram of an actively mode-locked OEO based on electric amplitude modulation. Continuous-wave (CW) light from a laser diode is fed into an electro-optic Mach-Zehnder modulator (MZM) biased at its quadrature point, where an optical variable attenuator is employed to finely control the power injected into the OEO cavity in order to obtain a stable mode-locking status. The intensity-modulated light passes through a section of single-mode fiber, and is converted to a microwave signal through a high-speed photodetector. The output signal from the photodetector is amplified by a low-noise amplifier to compensate for the power loss in the cavity, and is then filtered by a bandpass filter to select the oscillating modes. Different from a conventional OEO with a single-tone oscillation, an electric amplitude modulator is used in the OEO cavity to sinusoidally modulate the amplitude of the oscillating microwave signal before it reenters the MZM. If the modulation frequency $\Omega $ is equal to an integral multiple *N* of the cavity FSR $\Delta {f_{\textrm{FSR}}}$, i.e., $\Omega = N\Delta {f_{\textrm{FSR}}}$, the OEO works in a mode locking status ($N = 1$ for fundamental mode locking, and $N \ge 2$ for harmonic mode locking). In the time domain, the electric amplitude modulator acts as a periodic loss modulator with its modulation period $\Delta T = {1 / \Omega }$ equal to ${1 / N}$ of the OEO roundtrip time $\tau $. Only the signal at the time points with the lowest loss can be amplified to form a stable microwave pulse train. In the frequency domain, the electric amplitude modulator generates new sidebands with an identical phase. Therefore, the phase of the oscillating longitudinal modes in the net gain spectrum is locked, which is coherently superimposed in the time domain to form a microwave pulse train with its repetition frequency equal to the frequency of the externally-applied signal.

Mathematically, after a single-loop propagation in the OEO cavity, the voltage of the microwave signal injected into the electric amplitude modulator is calculated as

*B*is the voltage splitting ratio of the electric coupler (i.e., the ratio of the voltage feedback to the OEO loop to the input voltage). ${V_{ph}}$ is the output voltage from the photodetector, which is calculated as

*L*are the loss coefficient and the length of the single-mode fiber, respectively. ${G_A}$ is the gain coefficient of the low-noise amplifier, which can be expressed as where ${G_m}$, ${P_{sat}}$ and ${P_{in}}$ are the small-signal gain coefficient, the saturation power and the input power of the low-noise amplifier, respectively.

After modulation by a sinusoidal signal with a frequency equal to an integral multiple of the cavity FSR (i.e., $\Omega = N\Delta {f_{FSR}}$), the output microwave signal from the electric amplitude modulator is expressed as

*m*is the modulation index.

The kernel of the mode locking process is locking the phase of different longitudinal modes. Hence, the signal, the noise and the impulse response of the bandpass filter should be described in the complex plane. The microwave signal injected into the MZM can be expressed as

^{st}-order Bessel function of the first kind. In fact, the center frequency of the bandpass filter is close to ${f_c}$. Hence, the impulse response of the bandpass filter can be expressed as where $s(t )$ is the baseband impulse response. Therefore, by substituting Eqs. (6)–(8) into Eq. (4), it can be simplified as follows

The small-signal gain of the actively mode-locked OEO in the time domain is calculated as

In Eq. (9), the lowest loss occurs at $t = {k / \Omega }$, where *k* is an integer. Hence, in order to achieve mode locking oscillation, the small-signal gain at $t = {k / \Omega }$ should be larger than 0 dB, i.e.,

For center frequency of ${f_c}$, $\left|{\int_\tau {s(t )dt} } \right|= 1$. The oscillation threshold condition is simplified as

In [5], the threshold condition of a conventional OEO with a single-tone oscillation is given by

Therefore, it can be seen that the threshold gain of the actively mode-locked OEO is increased due to the electric amplitude modulation. Without this modulation (i.e., $m = 0$), the actively mode-locked OEO degenerates to a conventional OEO.## 3. Calculation method

An algorithm based on pulse tracking method is proposed to investigate the signal dynamics in the actively mode-locked OEO cavity. In a standard pulse tracking method, the initial input signal is a white noise, which propagates repeatedly in the cavity until reaching a stable status. Generally, the calculating time window is equal to the OEO roundtrip time $\tau $, which is so short that the phase noise cannot be characterized by the rough spectral resolution (the reciprocal of the time window). To break this limitation, the calculating time window in the proposed algorithm is expanded to $p\tau $, where *p* is an integer. Figure 2 presents the overflow of the proposed algorithm based on pulse tracking method. In each simulation cycle, the tracked pulse in a roundtrip time is delayed by $\tau $, and moves to the next roundtrip time. All the tracked pulses in different roundtrip time are stored, which is expressed as ${F_{\textrm{out}}}(t )$ shown in Fig. 2.

The output spectrum $FT(f )$ can be obtained through Fourier transform of ${F_{\textrm{out}}}(t )$ after reaching a stable status as

Hence, the phase noise of the actively mode-locked OEO can be easily calculated through the single-sideband power density as

In each single-cycle calculation, Eq. (8) is calculated in the time domain, which is a little complex since the output of the convolution process is affected by its input signal in the entire calculating time window. In general, the effective time duration of the filter impulse response $h(t )$ is much shorter than $\tau $. Hence, it is feasible to only consider the input signal with a short time duration of $2\tau $ in each convolution calculation. In the calculation, the roundtrip time $\tau $ is separated into the link delay ${\tau _1}$ and the filter group delay ${\tau _2}$, i.e., $\tau = {\tau _1} + {\tau _2}$. In the simulation, $s(t )$ is designed as the baseband impulse response of a Gaussian filter with a frequency response given by

*a*is a parameter determined by the 3-dB bandwidth of the bandpass filter $\Gamma = {{\sqrt {2\ln 2 \cdot a} } / \pi }$. The bandpass filter must be causal. Therefore, $s(t )$ is given by

$F_{\textrm{in}}^l$ and $F_{\textrm{out}}^l$ are defined as the complex envelopes of the tracked pulses at the input and the output of the bandpass filter in the ${l^{\textrm{th}}}$ simulation cycle, respectively. Figure 3 exhibits the computational steps for moving the tracked pulse from $F_{\textrm{in}}^l$ to $F_{\textrm{out}}^l$, i.e., a single-cycle calculation.

In Fig. 3, 2 convolution steps are carried out to maintain a precise delay. In the first convolution step, the input pulse is expressed as

$F_{\textrm{out}}^l$ is obtained by the convolution of $F_{\textrm{in},2\tau }^l$ and $s(t )$ defined in Eq. (16). Only the output $F_{\textrm{out},{\tau _1}}^l = F_{\textrm{out}}^l(t )$ in the range of $l\tau - {\tau _1} \le t < ({l + 1} )\tau $ is effective since the output in the range of $({l + 1} )\tau \le t < ({l + 1} )\tau + {\tau _2}$ is affected by the filter input in the range of $({l + 1} )\tau \le t < ({l + 1} )\tau + {\tau _2}$, which is a fraction of $F_{\textrm{in}}^{l + 1}$ in the next cycle. Then, $F_{\textrm{out},{\tau _1}}^l$ propagates around the loop, and transforms into $F_{\textrm{in},{\tau _1}}^{l + 1}$ in the range of $({l + 1} )\tau \le t < ({l + 1} )\tau + {\tau _1}$, which is calculated by Eq. (8) as

*x*. In the second convolution step, $F_{\textrm{in},{\tau _1}}^{l + 1}$ is combined with $F_{\textrm{in},2\tau }^l$ as the input signal, which is given by

Through the convolution of $F_{\textrm{in},2\tau + {\tau _1}}^l$ and $s(t )$, the effective output is $F_{\textrm{out}}^l$ in the range of $l\tau + {\tau _2} \le t < ({l + 1} )\tau + {\tau _2}$, which is the entire output pulse of the bandpass filter in the ${l^{\textrm{th}}}$ cycle. Then, $F_{\textrm{out}}^l$ propagates around the loop to evolve into $F_{\textrm{in}}^{l + 1}$ in the range of $({l + 1} )\tau \le t < ({l + 2} )\tau $, which is calculated as

Hence, $F_{\textrm{in}}^l$ is moved to the next roundtrip time precisely as $F_{\textrm{in}}^{l + 1}$, and $F_{\textrm{in}}^{l + 1}$ is combined with $F_{\textrm{in}}^l$ as the input signal $F_{\textrm{in},2\tau }^{l + 1}$ in the next single-cycle calculation. The output waveform $F_{\textrm{out}}^l$ in each time window is stored to form the output signal ${F_{\textrm{out}}}(t )$.

## 4. Simulation results

By using the proposed theoretical model and calculation method, a numerical simulation is carried out to characterize an actively mode-locked OEO based on electric amplitude modulation. In the simulation, the roundtrip time $\tau $ is 10.05 µs, where ${\tau _1}$ is 10 µs (corresponding to a single-mode fiber with a length of $L = 2\textrm{ km}$ and a loss coefficient of $\alpha = 0.25\textrm{ dB/km}$), and ${\tau _2}$ is 0.05 µs. The electric amplitude modulator is driven by a sinusoidal signal with a frequency of 99.502 kHz to achieve fundamental mode locking, where the modulation index is $m = 0.67$. It should be pointed out that this model is also feasible to simulate a harmonically mode-locked OEO by simply setting the modulation frequency $\Omega $ to be an integral multiple *N* of the cavity FSR $\Delta {f_{\textrm{FSR}}}$, i.e., $\Omega = N\Delta {f_{\textrm{FSR}}}\textrm{ }({N \ge 2} )$. The center frequency and the 3-dB bandwidth of the bandpass filter are set to be 2 GHz and 70 MHz, respectively. The power of the CW light at 1560 nm from the laser diode is 30 mW. In addition, the power spectral density of the noise is set to be −160 dBm/Hz. The device parameters in the OEO cavity are listed in Table 1.

In order to obtain the temporal waveform containing the carrier information, the number of grid points in a single roundtrip time is set to be 100,000 to guarantee a sufficient bandwidth of 10 GHz. The small-signal gain is set to be 1.006 by properly controlling the transmissivity ${\sigma _{att}}$. The simulation is self-starting from a white noise, and typically converges after 1500 roundtrips. Figure 4 presents the output temporal waveform, where a microwave pulse train with a peak voltage of 0.37 V and a pulse width (i.e., full width at half maximum) of 0.38 µs is generated. The repetition rate and the carrier frequency of the microwave pulse train are 99.502 kHz and 2.00 GHz, respectively, which fit in with the theoretical results.

To obtain the output spectrum, the number of grid points in a single roundtrip time is reduced to 4000 to save computation consumption, which has no influence on the spectral resolution if the time window used to achieve Fourier transform calculation is enough. After 12,000 roundtrips, the results in the last 4000 cycles are stored to calculate the output spectrum based on Eq. (13), where the frequency resolution is 24.875 Hz. Figure 5 shows the output spectrum. The 3-dB bandwidth and the mode interval are 2.41 MHz and 99.502 kHz, respectively. It can be seen that there are some notches on the envelope of the output spectrum, which is attributed to that the output pulse is not a Gaussian pulse, but closer to a super Gaussian one expressed as $|{A(t )} |= A\textrm{exp} ({ - {{|{{t / {{T_0}}}} |}^k}} )$ ($k$ and ${T_0}$ denote the order and the half width of the microwave pulse, respectively). Figure 6 presents the envelopes of the microwave pulse and the spectrum together with their super Gaussian fitting curves, where *k* and ${T_0}$ are equal to 3.22 and 0.273 µs, respectively. It should be pointed out that mode locking can be achieved even if there is a small frequency detuning $\Delta f = \Omega - {1 / T}$. Simulation results indicate that, under the above-mentioned cavity parameters, a stable mode locking status can be realized in the range of $- 4.5\textrm{ Hz} < \Delta f < 4.5\textrm{ Hz}$. In fact, the allowable frequency detuning range to achieve a stable mode locking status is mainly determined by the quality factor *Q* of the OEO cavity. For a higher *Q*, the linewidth of each longitudinal mode in the OEO cavity is smaller. In such a case, the allowable frequency detuning range to achieve a stable mode locking status will further decrease.

Figure 7 exhibits the phase noise of the actively mode-locked OEO (blue line) and a conventional OEO with a single-tone oscillation (red line), where the cavity parameters of the conventional OEO are identical to those of the actively mode-locked OEO. It can be seen that the phase noise of the actively mode-locked OEO at 100-Hz frequency offset is 33.6 dB lower than that in a conventional OEO. Although the exact phase noise level at different frequency offset in Fig. 7 may slightly vary in different simulation due to the randomly added white noise, the great close-to-carrier phase noise optimization is not attributed to this random variation but attributed to the mode locking technology.

## 5. Experimental results

An experiment is carried out to demonstrate the proposed theoretical model and the numerical simulation results. In the experiment, a distributed feedback laser diode generates a CW light with a power of 16 dBm and a center wavelength of 1560 nm. After attenuated by an optical variable attenuator, the CW light is intensity-modulated by a 20 Gb/s electro-optic MZM (EOSPACE) biased at its quadrature point. Then, it passes through a spool of single-mode fiber (YOFC) with a length of 2.05 km. A 20 Gb/s photodetector (HP 11982A) is employed to convert the intensity-modulated light into the microwave signal, which is then amplified by a low-noise amplifier (Qotana) with an operation frequency range from 1 GHz to 20 GHz and a gain of 25 dB, and filtered by a bandpass filter with a center frequency of 2 GHz and a 3-dB bandwidth of 70 MHz. An electric power divider (GTPD COMB50G) is used to output the generated microwave signal. Fundamental mode locking is realized by modulating the amplitude of the microwave signal in the cavity through an electric signal modulator (HP 11665B) driven by a sinusoidal signal with a frequency of a 97.57 kHz from a function generator (Hantek HDG2022B). The temporal waveform and the spectrum of the generated microwave signal are measured by an electrical spectrum analyzer (R&S FSU50, 20 Hz-50 GHz) and a high-speed real-time oscilloscope (Tektronix DPO75002SX, 100 GS/s, 33 GHz), respectively.

Figure 8 shows the measured temporal waveform of the generated microwave pulse train. It can be seen that the experimental result in Fig. 8 matches with the simulation one in Fig. 4. The pulse width is measured to be 0.792 µs, which is larger than the simulation result. This is attributed to the inaccurate gain adjustment by using a mechanical optical variable attenuator in the experiment. Therefore, the mode locking status in the experiment is poorer than that in the simulation. The poor mode locking status due to the improper gain can be briefly explained as follows. In the time domain, as the small-signal gain increases, signals in a wider time interval with a net gain larger than 0 dB can build up from noise, which gives rise to a larger pulse width. In the frequency domain, the increase of gain allows more side modes to generate from the initial noise with random phase, which deteriorates the mode locking status. Figure 9 presents the measured spectrum of the generated microwave pulse train, where the 3-dB bandwidth and the mode interval are 1.05 MHz and 97.57 kHz, respectively. Compared with the simulation result, the relatively narrow bandwidth in the experiment is also an evidence of unideal mode locking status. In addition, the notches on the envelope of the output spectrum indicate that the generated microwave pulse is a super Gaussian one. Figure 10 exhibits the envelopes of the microwave pulse and the spectrum together with their super Gaussian fitting curves, where *k* and ${T_0}$ are equal to 7.81 and 0. 479 µs, respectively.

Figure 11 shows the phase noise of the actively mode-locked OEO (blue line) and a conventional OEO with a single-tone oscillation (red line), which is measured by using the phase noise module of the electrical spectrum analyzer. It can be seen that the phase noise of the actively mode-locked OEO at 100-Hz frequency offset is 28.6 dB lower than that in a conventional OEO, which agrees with the simulation result in Fig. 7. Both the simulation and experimental results indicate that the close-to-carrier phase noise of the actively mode-locked OEO is much smaller than that of a conventional OEO. Finally, it should be pointed out that the disagreement beyond frequency offset of 300 Hz compared with the simulation results in Fig. 7 is attributed to the dynamic range of the electrical spectrum analyzer. The power of each mode in the spectrum of the generated microwave pulse train is much smaller than that of the single-tone oscillation. Therefore, in the phase noise measurement by using an electrical spectrum analyzer, the measurement dynamic range puts a limit to the measurement sensitivity in the far frequency offset range, which is actually with a lower phase noise. The low close-to-carrier phase noise of the actively mode-locked OEO is favorable for realizing a pulse Doppler radar with a high velocity detection sensitivity.

## 6. Conclusion

In summary, a theoretical model and its calculation method are proposed to simulate an actively mode-locked OEO based on electric amplitude modulation. Based on the model, the waveform, the spectrum and the phase noise characteristic of the generated microwave pulse train from an actively mode-locked OEO are obtained by numerical simulation, where the results fit in with the experimental results. The proposed model can be used to design an actively mode-locked OEO based on electric amplitude modulation, and study the dynamic process in it.

## Funding

National Key Research and Development Program of China (2019YFB2203800); National Natural Science Foundation of China (61421002, 61927821); Fundamental Research Funds for the Central Universities (ZYGX2020ZB012).

## 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.

## References

**1. **R. Prasad, “Overview of wireless personal communications: microwave perspectives,” IEEE Commun. Mag. **35**(4), 104–108 (1997). [CrossRef]

**2. **Z. Y. Peng and C. Z. Li, “Portable Microwave Radar Systems for Short-Range Localization and Life Tracking: A Review,” Sensors **19**(5), 1136 (2019). [CrossRef]

**3. **W. J. Plant, W. C. Keller, and K. Hayes, “Measurement of River Surface Currents with Coherent Microwave Systems,” IEEE T. Geosci. Remote. Sens. **43**(6), 1242–1257 (2005). [CrossRef]

**4. **R. A. Minasian, E. H. W. Chan, and X. Yi, “Microwave photonic signal processing,” Opt. Express **21**(19), 22918–22936 (2013). [CrossRef]

**5. **X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B **13**(8), 1725–1735 (1996). [CrossRef]

**6. **D. Eliyahu, D. Seidel, and L. Maleki, “Phase Noise of a High Performance OEO and an Ultra Low Noise Floor Cross-Correlation Microwave Photonic Homodyne System,” in IEEE International Frequency Control Symposium (2008), pp. 811–814.

**7. **T. F. Hao, J. Tian, D. Domenech, W. Li, N. H. Zhu, J. Capmany, and M. Li, “Toward Monolithic Integration of OEOs: From Systems to Chips,” J. Lightwave Technol. **36**(19), 4565–4582 (2018). [CrossRef]

**8. **E. C. Levy and M. Horowitz, “Single-cycle radio-frequency pulse generation by an optoelectronic oscillator,” Opt. Express **19**(18), 17599–17608 (2011). [CrossRef]

**9. **A. Sherman and M. Horowitz, “Ultralow-repetition-rate pulses with ultralow jitter generated by passive mode-locking of an optoelectronic oscillator,” J. Opt. Soc. Am. B **30**(11), 2980–2983 (2013). [CrossRef]

**10. **Z. Zeng, L. J. Zhang, Y. W. Zhang, H. Tian, Z. Y. Zhang, S. J. Zhang, H. P. Li, and Y. Liu, ““Microwave pulse generation via employing an electric signal modulator to achieve time-domain mode locking in an optoelectronic oscillator,” Opt. Lett. **46**(9), 2107–2110 (2021). [CrossRef]

**11. **Y. K. Chembo, L. Larger, H. Tavernier, R. Bendoula, E. Rubiola, and P. Colet, “Dynamic instabilities of microwaves generated with optoelectronic oscillators,” Opt. Lett. **32**(17), 2571–2573 (2007). [CrossRef]

**12. **E. C. Levy, M. Horowitz, and C. R. Menyuk, “Modeling optoelectronic oscillators,” J. Opt. Soc. Am. B **26**(1), 148–159 (2009). [CrossRef]

**13. **E. C. Levy and M. Horowitz, “Theoretical and experimental study of passive mode-locked optoelectronic oscillators,” J. Opt. Soc. Am. B **30**(1), 107–112 (2013). [CrossRef]