Time-domain convolution model for studying oscillation dynamics in an injection-locked optoelectronic oscillator

Weichen Yuan; Weichen Yuan; Zhenwei Fu; Zhenwei Fu; Yilin Wu; Yilin Wu; Di Peng; Lingjie Zhang; Lingjie Zhang; Zhen Zeng; Zhen Zeng; Yali Zhang; Yali Zhang; Zhiyao Zhang; Zhiyao Zhang; Shangjian Zhang; Shangjian Zhang; Yong Liu; Yong Liu

doi:10.1364/OE.473268

1. Introduction

High-performance microwave oscillators are urgently needed in the advanced radar systems and the sixth-generation (6G) wireless communications [1,2]. Nevertheless, the conventional microwave oscillators based on electronic technology encounter difficulties in generating high-frequency signals with low phase noise. To circumvent this obstacle, optoelectronic oscillators (OEOs) have been widely studied in the past two decades, which are recognized as a promising candidate to generate microwave signals with ultra-low phase noise in a broadband range [3–7]. The most annoying problem in an OEO lies in that the small mode interval induced by the large time delay in the cavity generally prohibits it from oscillating at a pure single frequency. To date, various techniques have been proposed to realize single-mode oscillation, such as dual-loop architecture [8–10], injection locking [11–13] and frequency conversion filtering [14]. Thereinto, injection locking is an efficient solution to suppress the unwanted modes in the OEO with an ultra-high Q-factor. Based on this technique, single-mode oscillation with a side-mode suppression ratio (SMSR) as high as 80 dB has been demonstrated in the OEO [13].

In an injection-locked OEO, aside from the injection locking effect, rich dynamic phenomena such as frequency pulling and asymmetrical spectrum generation have also been observed in the experiment when the frequency of the externally-injected microwave signal deviates from that of the longitudinal mode [15,16]. A well-established model is of great importance to achieve an in-depth study of these dynamic processes. The first theoretical model for an OEO was proposed by X. S. Yao and L. Maleki in 1996 based on the quasi-linear theory [3]. Although this model can only be used to predict the frequency-domain characteristic of a free-running OEO, it lays a firm foundation for the theoretical research of various advanced OEOs. Based on this model, a modified quasi-linear approach was proposed to study the temporal evolution of the oscillation signal in a single-loop OEO [17]. Subsequently, many efforts were made to model an injection-locked OEO based on the previous studies of the injection locking effect in a conventional microwave oscillator [11,14,18]. However, these theoretical models can only be applied to analyze the phase noise characteristic of the injection-locked OEO when the frequency of the externally-injected microwave signal is identical to that of a longitudinal mode in the cavity. In 2021, Z. Q. Fan et al. successfully explained the injection locking and the frequency pulling phenomena in the injection-locked OEO based on R. Adler’s work and A. Banerjee’s work [16,18,19]. Nevertheless, this work has not explained the shifting of the asymmetrical spectrum in each oscillation cycle when the OEO works in the quasi-locked state. Recently, M. Hasan et al. proposed a theoretical model to explain the essential features of a large delay OEO under different injection circumstances [20]. This work is beneficial for the in-depth study of the injection-locked OEO. However, a model capable of revealing the oscillation dynamics including the building-up process in an injection-locked OEO is still absent.

In this paper, a time-domain convolution model is proposed to study the oscillation dynamics in an injection-locked OEO. Based on this model, the real-time dynamic process in the injection-locked OEO can be numerically simulated, which is favorable for providing an insight into the oscillation dynamics intuitively. By using this model, the injection locking, the frequency pulling and the asymmetrical spectrum generation phenomena are numerically studied, where the simulation results are consistent with the experimental results.

2. Theoretical model

Figure 1 shows the architecture of the injection-locked OEO. Continuous-wave (CW) light from a laser diode (LD) is injected into a Mach-Zehnder Modulator (MZM) biased at its quadrature point, where the optical power is modulated by the feedback microwave signal in the OEO loop. After modulation, the optical signal propagates through a spool of single-mode fiber (SMF), and is then detected by using a photodetector (PD). The microwave signal from the PD is amplified, filtered, and coupled with an externally-injected microwave signal before reentering the MZM. The optical variable attenuator (OVA) between the LD and the MZM is used to finely adjust the optical power injected into the OEO cavity, which ensures a stable oscillation. When the frequency of the injected signal is equal to that of a longitudinal mode in the OEO, injection locking is achieved, where single-mode oscillation with a high SMSR is realized. When the frequency of the injected signal deviates from that of any longitudinal mode in the OEO by a small value, frequency pulling occurs, where all longitudinal modes have an identical frequency shift. When the frequency of the injected signal deviates from that of any longitudinal mode in the OEO by a large value, the OEO enters an unstable injection-locking state, where asymmetrical spectrum appears.

Fig. 1. Architecture of the injection-locked OEO. LD: laser diode; OVA: optical variable attenuator; MZM: Mach-Zehnder modulator; SMF: single-mode fiber, PD: photodetector; LNA: low-noise amplifier; BPF: bandpass filter; EC: electrical coupler; MS: microwave source.

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For the injection-locked OEO, a time-domain convolution model is established to study the real-time oscillation process. The optical power from the MZM is expressed as

(1)$$P(t )= ({{{{\alpha_L}{\sigma_L}{P_0}} / 2}} )\cdot \{{1 - \eta \sin ({{{\pi {V_{in}}(t )} / {{V_\pi } + {{\pi {V_B}} / {{V_{\pi 0}}}}}}} )} \},$$

where α_L and σ_L are the optical power transmittance of the MZM and the OVA, respectively. P₀ is the optical power from the LD. V_in (t) is the voltage of the microwave signal injected into the MZM. V_B is the bias voltage applied to the MZM. V_π and V_π₀ are the radio-frequency (RF) and the direct-current (DC) half-wave voltages of the MZM, respectively. η is a parameter determined by the extinction ratio of the MZM, where the extinction ratio is (1+η)/(1-η).

After being detected by using a PD, the optical signal is converted into a microwave signal. Then, the microwave signal passes through a low-noise amplifier (LNA), a bandpass filter (BPF) and an electrical coupler (i.e., EC1 in Fig. 1) in sequence. The microwave signal from EC1 is calculated as

(2)$${V_{out}}(t )= {\beta _1}\{{[{\rho RP(t )\textrm{exp} ({ - \alpha L} )+ {n_{PD}}(t )} ]{G_A} + {n_{LNA}}(t )} \}\ast {h_{BPF}}(t ),$$

where β₁ is related to the power splitting ratio of EC1. ρ and R are the responsivity and the output impedance of the PD, respectively. G_A is the voltage gain of the LNA. The exponential term exp(-αL) represents the optical power loss induced by the SMF, where α and L are the loss coefficient and the length of the SMF, respectively. n_PD(t) and n_LNA(t) are noises introduced by the PD and the LNA, respectively. The symbol * denotes the convolution operation. h_BPF (t) is the unit impulse response of the BPF.

The microwave signal is then coupled with the injected microwave signal via another electrical coupler (i.e., EC2 in Fig. 1), and acts as the input signal of the MZM for the next oscillation cycle. Hence, the input signal of the MZM is calculated as

(3)$${V_{in}}({t - \tau } )= {\beta _2}{V_{out}}(t )+ {\beta _3}{V_i}(t )= {\beta _2}{V_{out}}(t )+ {\beta _3}|{{V_i}(t )} |\textrm{exp} ({ - j2\pi {f_i}t + j{\varphi_i}(t )} ),$$

where β₂ and β₃ are related to the power splitting ratio of EC2. τ is the time delay of the OEO loop. f_i, φ_i(t) and |V_i(t)| are the frequency, the initial phase and the voltage amplitude of the injected microwave signal V_i(t).

The time-domain convolution model shown in Eqs. (1–3) has no analytic solution in most cases. Hence, numerical calculation based on pulse tracking method as depicted in Fig. 2 is an efficient method to solve this model, which can provide a deep insight into the oscillation dynamics. In the numerical simulation, the time window is set to be equal to the time delay of the OEO loop (i.e., τ in Fig. 2), which mainly involves the propagation delay of the SMF and the group delay of the BPF.

Fig. 2. Schematic diagram of the numerical simulation based on pulse tracking method to study the real-time dynamic process in the injection-locked OEO.

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For simplicity, the BPF is supposed to be with a Gaussian-shape passband. Hence, the unit impulse response is written as

(4)$${h_{BPF}}(t )= \left\{ {\sqrt {{a / \pi }} \textrm{exp} ({ - a{{({t - {\tau_l}} )}^2}} )\cdot \textrm{exp} ({ - j2\pi {f_c}({t - {\tau_l}} )} )} \right\}u(t ),$$

where a is a parameter determined by the 3-dB bandwidth of the BPF, and the bandwidth is calculated as π⁻¹ (2aln2)^1/2. f_c and τ_l are the center frequency and the group delay of the BPF, respectively. u(t) is the unit step function. In the numerical simulation, the limited time window and the relatively long impulse response time of the BPF may introduce a truncation error into the convolution computation in Eq. (2). Hence, the two-step calculation method in [21] is adopted.

In each roundtrip of the OEO, the injected microwave signal and the additive noises from the active devices contribute to the building-up process of the oscillation signal. The additive noises include the thermal noise, the shot noise, the relative intensity noise and the noise introduced by the LNA. Thereinto, the first three kinds of noise contribute to the noise from the PD in Eq. (2), i.e., n_PD (t). The root mean square (RMS) value of n_PD (t) can be calculated as

(5)$$\sqrt {\left\langle {n_{_{PD}}^2(t )} \right\rangle } = \sqrt {({kT + 2eIR + RIN \cdot {I^2}R} )RB} ,$$

where k, T and e are the Boltzmann constant, the Kelvin temperature and the elementary charge, respectively. I is the average current from the PD. RIN is the relative intensity noise. B is the bandwidth. In addition, the RMS value of the noise n_LNA (t) introduced by the LNA in Eq. (2) can be calculated as

(6)$$\sqrt {\left\langle {n_{_{LNA}}^2(t )} \right\rangle } = {G_A}\sqrt {({NF - 1} )kTRB} ,$$

where NF is the noise factor of the LNA.

As for the initial phase φ_i(t) in Eq. (3), there are two models available. The first one considers φ_i(t) to satisfy a complex Gaussian distribution, which is widely used in the previous study of the injection-locked OEO. In this condition, the phase noise of the injected microwave signal is a white noise. The second model is obtained by measuring the injected single-tone microwave signal via a high-speed real-time oscilloscope. The instantaneous phase of the measured single-tone microwave signal can be calculated through Hilbert transform as

(7)$$\Phi (t )= \arctan \left( {\frac{{Re ({{V_i}(t )} )\ast {1 / {\pi t}}}}{{Re ({{V_i}(t )} )}}} \right) = 2\pi {f_i}t + {\varphi _i}(t ).$$

Hence, the initial phase of the injected single-tone microwave signal can be calculated as

(8)$${\varphi _i}(t )= \Phi (t )- 2\pi {f_i}t.$$

After the OEO reaches the stable oscillation state (i.e., the N^th cycle in Fig. 2), numerous roundtrips of the oscillation signal (i.e., from the N^th circle to the (N + n)^th cycle) are stored in sequence as V_out (t), which is used to calculate the spectrum and the phase noise of the generated microwave signal with a high spectral resolution. Thereinto, the spectrum of the generated signal from the OEO is obtained by implementing Fourier transformation of V_out (t) as

(9)$${V_{out}}(t )\textrm{exp} ({j2\pi {f_c}t} )\buildrel {\cal F} \over \longleftrightarrow {F_{out}}(f ).$$

In addition, the phase noise, i.e., the ratio of the power density at an offset frequency from the carrier to the total power of the carrier signal, is calculated as

(10)$$L({\Delta f} )= \frac{{n\tau {{|{F({\Delta f} )} |}^2}}}{{2\sum\limits_f {{{|{{F_{out}}(f )} |}^2}} }},$$

where Δf = f - f_c represents the frequency offset from the oscillation frequency of the injection-locked OEO. n is the number of the cycles stored in V_out (t). Therefore, the spectral resolution is equal to 1/nτ.

3. Results and discussions

Based on the time-domain convolution model, the oscillation dynamics in the injection-locked OEO is studied, and the results are compared with those obtained by the experiment. In the experiment, a LD at 1548.93 nm provides the CW light with a power of 16.73 dBm. The CW light is then modulated by the feedback microwave signal via a 20 Gb/s MZM (EOSPACE) biased at its quadrature-point. After propagating through a 2.05-km-long SMF, the optical signal is detected by using a 20 Gb/s PD (HP 11982A). The microwave signal from the PD is amplified by using a LNA (Qotana) with a small-signal gain of 20 dB and an operation bandwidth from 1 GHz to 20 GHz, and then filtered by using a BPF with a center frequency of 3 GHz and a 3-dB bandwidth of 52 MHz. Two 6-dB ECs are used to inject the signal from a microwave signal generator (R&S SMB100A, 100 kHz−12.75 GHz) into the OEO loop, and to extract the generated signal to a spectrum analyzer (R&S RTO1024, 20 Hz-50 GHz) and a phase noise analyzer (R&S FSWP, 1 MHz-50 GHz) for measurement. In the numerical simulation, the total time delay of the OEO loop is 10.2 µs, which is composed of 10-µs time delay in a 2-km-long SMF and 0.2-µs group delay in the BPF. The sampling frequency is set to be 400 MHz, which corresponds to a temporal resolution of 2.5 ns. The center frequency and the 3-dB bandwidth of the BPF are 3 GHz and 52 MHz, respectively. The CW light at 1550 nm from the LD is with a power of 16.5 dBm. Other parameters of the devices used in the OEO loop are listed in Table 1. It should be pointed out that, although η = 1 corresponds to an infinite extinction ratio of the MZM in the numerical simulation, it has a negligible influence on the simulation accuracy, since the extinction ratio of the commercial MZM is generally larger than 20 dB. On this condition, the modulation depth for an infinite extinction ratio is nearly identical to that for an extinction ratio larger than 20 dB.

Table 1. Parameters of the devices in the OEO for numerical simulation

View Table

Firstly, the frequency of the injected signal is set to be identical to that of a longitudinal mode. Figure 3 and Fig. 4 exhibit the simulated and the measured output spectra of the OEO under different injection power, respectively. Thereinto, Fig. 3(a) and Fig. 4(a) present the output spectra of the free-running OEO without an injected signal. The abscissa in Fig. 3 and Fig. 4 is the frequency offset from 3 GHz. The SMSR increases with the injection power as depicted in Figs. 3(b) and (c), which fits in with the experimental results in Figs. 4(b) and (c). In addition, the power of the generated microwave signal increases with the injection power, indicating that the injected signal also contributes to the output signal since the PD works in the linear region. The different noise floors in Fig. 3 and Fig. 4 are attributed to the different spectral resolution bandwidth (RBW), i.e., 9.8039 Hz in the simulation and 500 Hz in the experiment. These results indicate that the time-domain convolution model can be used to accurately simulate the side-mode suppression effect in the injection-locked OEO.

Fig. 3. Simulated output spectra of the OEO (a) in the free-running state, (b) with an injection power of −20 dBm, and (c) with an injection power of −10dBm.

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Fig. 4. Measured output spectra of the OEO (a) in the free-running state, (b) with an injection power of −20 dBm, and (c) with an injection power of −10dBm.

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Figures 5(a) and (b) present the simulated phase noise of the generated microwave signal under different injection power, which are obtained by using the white noise model and the measurement model for the initial phase of the injected signal, respectively. The spurs in the phase noise curve of the injected signal in Fig. 5(b) are attributed to the near-frequency stray components. It can be seen from Fig. 5 that the near-end phase noise of the generated microwave signal is determined by that of the injected signal, and shows no obvious change as the injection power increases. The far-end phase noise is determined by that of the free-running OEO, and shows a gradual deterioration as the injection power increases. These conclusions fit in with the measurement results in Fig. 6, indicating that the time-domain convolution model can be used to accurately calculate the phase noise performance of the generated microwave signal in the injection-locked OEO.

Fig. 5. Simulated phase noise of the generated microwave signal under different injection power obtained by using (a) the white noise model and (b) the measurement model for the initial phase of the injected signal.

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Fig. 6. Measured phase noise of the generated microwave signal under different injection power.

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Then, the frequency of the injected signal is set to be with a small deviation from that of the longitudinal mode. According to the theory in [16], the locking range is calculated to be ±1.25 kHz when the injection power is −20 dBm. Figure 7 shows the simulated output spectrum of the OEO when the injected signal is with a power of −20 dBm and a frequency deviation of 1 kHz (i.e., the frequency of the injected signal is 1 kHz larger than that of the longitudinal mode). The abscissa in Fig. 7 is the frequency offset from 3 GHz. It can be seen from Fig. 7 that, since the frequency deviation of the injected signal is within the locking range, the side-mode suppression is still effective. In addition, all of the longitudinal modes have an identical frequency shift of 1 kHz, which fits in with the measurement result in Fig. 8. These results indicate that the frequency pulling phenomenon in the injection-locked OEO can be simulated by using the time-domain convolution model.

Fig. 7. Simulated output spectra of the OEO when the injected signal is with a power of −20 dBm and a frequency deviation of 1 kHz.

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Fig. 8. Measured output spectra of the OEO when the injected signal is with a power of −20 dBm and a frequency deviation of 1 kHz.

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Subsequently, the frequency of the injected signal is set to be with a large deviation from that of the longitudinal mode. Figures 9(a) and (b) present the simulated output spectra of the OEO when the injected signal at the −20 dBm level is with a frequency deviation of 10 kHz and −10 kHz, respectively. The abscissa in Fig. 9 is the frequency offset from 3 GHz. It can be seen from Fig. 9 that, since the frequency deviation of the injected signal is out of the locking range, the side-mode suppression effect disappears, and the asymmetrical spectrum generation phenomenon occurs. A series of stair-step sidebands emerge around the primary oscillation modes as depicted by the orange dashed lines in Fig. 9. When the frequency deviation of the injected signal is 10 kHz, the stair-step sidebands appear on the low-frequency side of the primary oscillation modes. When the frequency deviation of the injected signal is −10 kHz, the stair-step sidebands appear on the high-frequency side of the primary oscillation modes. In both cases, the frequency interval between the sidebands and the corresponding oscillation modes is equal to 10 kHz. Figure 10 exhibits the measurement results, which verifies the correctness of the simulation results. These results indicate that the time-domain convolution model can be used to accurately simulate the asymmetrical spectrum generation phenomenon in the injection-locked OEO.

Fig. 9. Simulated output spectra of the OEO when the injected signal at the −20 dBm level is with a frequency deviation of (a) 10 kHz and (b) −10 kHz.

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Fig. 10. Measured output spectra of the OEO when the injected signal at the −20 dBm level is with a frequency deviation of (a) 10 kHz and (b) −10 kHz.

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Figures 11(a) and (b) show the simulated output spectra of the OEO when the frequency deviation of the injected signal is equal to half of the free spectral range (FSR) and an FSR (i.e., the frequency of the injected signal is equal to that of another longitudinal mode), respectively. Figures 12(a) and (b) exhibit the corresponding measurement results. The abscissa in Fig. 11 and Fig. 12 is the frequency offset from 3 GHz. These results indicate that the time-domain convolution model can be used to accurately simulate the oscillation dynamics even under a large frequency deviation.

Fig. 11. Simulated output spectra of the OEO when the frequency deviation of the injected signal is equal to (a) half of the FSR and (b) an FSR.

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Fig. 12. Measured output spectra of the OEO when the frequency deviation of the injected signal is equal to (a) half of the FSR and (b) an FSR.

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Finally, it should be pointed out that the proposed model describes the oscillation process in the time domain. Hence, the prominent advantage lies in that it can be used to simulate the building-up process of the oscillation signal in the OEO. Figures 13(a) and (b) exhibit the simulated spectral building-up processes of the OEO when the frequency of the injected signal is identical to that of the longitudinal mode and with a deviation of 10 kHz, respectively. In addition, Fig. 13(c) shows the simulated spectral building-up process of the free-running OEO. It should be pointed out that the rough frequency resolution in Fig. 13 is attributed to the limited time window of each roundtrip in the fast Fourier transform (FFT) calculation. The most interesting thing is that the sidebands appear alternately on both sides of the primary oscillation mode as depicted in the inset of Fig. 13(b). This phenomenon has not been reported before. Therefore, the proposed time-domain convolution model is a powerful tool to study the oscillation dynamics in the injection-locked OEO.

Fig. 13. Simulated spectral building-up processes of the OEO when (a) the frequency of the injected signal is identical to that of the longitudinal mode, (b) the frequency of the injected signal is with a deviation of 10 kHz, and (c) there is no injection signal.

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4. Conclusion

In summary, we have proposed a time-domain convolution model for studying the oscillation dynamics in the injection-locked OEO. Based on the model, the injection locking, the frequency pulling and the asymmetrical spectrum generation phenomena have been numerically simulated, where the simulation results fit in with the experimental results. The model has also been used to simulate the building-up process of the oscillation signal in the injection-locked OEO. To the best knowledge of the authors, the alternating appearance of the sidebands on both sides of the primary oscillation mode is reported for the first time when the frequency of the injected signal is with a large deviation from that of the longitudinal mode. In addition, the proposed model can be used to reveal the relations between the properties of the oscillation signal and the parameters of the devices. Hence, it is a powerful tool to design an injection-locked OEO, and study the dynamic process in it.

Funding

National Key Research and Development Program of China (2018YFE0201900); National Natural Science Foundation of China (61927821, 62175038); 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.

Supplemental document

See Supplement 1 for supporting content.

References

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Parameter	Symbol	Value	Parameter	Symbol	Value
Half-wave voltage of the MZM	V_π	5 V	Parameter related to extinction ratio	η	1
Loss coefficient of the SMF	α	0.2 dB/km	Responsivity of the PD	ρ	0.8 A/W
Output impendence of the PD	R	50 Ω	Voltage gain	G_A	10
Noise figure of the LNA	NF	5 dB	Boltzmann constant	k	1.380649 J/K
temperature	T	298 K	Elementary charge	e	1.6 × 10⁻¹⁹ C
Relative intensity noise of the LD	RIN	−140 dBc/Hz	Power splitting ratio of EC1	β₁	0.5
Power splitting ratio of EC2	β₂	0.5	Power splitting ratio of EC3	β₃	0.5

Time-domain convolution model for studying oscillation dynamics in an injection-locked optoelectronic oscillator

Abstract

1. Introduction

2. Theoretical model

3. Results and discussions

4. Conclusion

Funding

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (13)

Tables (1)

Equations (10)

Optics Express