1. Introduction

Optoelectronic oscillators (OEOs) were first reported by Yao and Maleki in 1996 [1–3], which is a closed feedback loop that is formed with two concatenated optical and electronic branches [4,5], and has been widely employed to generate ultrapure high-frequency microwaves [3,6–9]. The model Yao and Maleki employed for studying OEO is a quasi-linear model [1], which can be considered as a modification of the linear model for analyzing linear feedback system [10,11]. Both the linear model and quasi-linear model consider the final output of a feedback system as the summation of all recurrence circulating fields [1,2]. The difference is that the homogeneity is satisfied for a linear feedback system but is not satisfied for the OEO. The Yao-Maleki quasi-linear model indicates that the closed-loop gain decreases from small signal gain to unity with the increasing of the oscillating amplitude [1,2], which ensures stability of the OEO. Quasi-linear model is good enough for OEOs under stable state, however, it does not apply to analyze the nonlinear dynamics such as bifurcation phenomena [12,13]. Resolving nonlinear dynamics is highly important for generation of ultrapure microwaves [12–17], other OEO based applications such as senses and measurement systems [4,5,18], and secure chaotic communications [19,20]. Thus, OEOs require other analysis methods in addition to the quasi-linear model.

A popular method for resolving dynamic instability problems is the delay-differential model proposed by Y.k.Chembo [12,13]. The delay-differential model has successfully predicted that the bifurcation phenomena show up when the open-loop small-signal gain is larger than a critical threshold value of 2.31 [12,13]. However, the provided bifurcation diagrams and the threshold value of 2.31 apply to the class of OEOs considering only nonlinear effects of modulator. For a practical OEO structure, the nonlinear properties can be complex due to the saturation characteristics of employed components in addition to modulator such as RF electronic amplifiers (EAs), optical amplifiers (OAs) and photodetectors (PDs). The OEOs with complex nonlinear properties can maintain a stable state even the small-signal gain beyond the threshold value of 2.31. Furthermore, the delay-differential model only considering the stability as a function of small-signal gain. However, for the class of OEOs with an externally injected microwave [21–24], which is a popular architecture for improving oscillation frequency stability [25–28], the injected amplitude is another variable and is deserved to be discussed.

Iterated map is a common graphical method for analyzing nonlinear iteration problems [29–32], and it can be employed to resolving the nonlinear dynamics in time-delay feedback system, such as the voltage evolution in an optoelectronic feedback system with time delay [19,29,33]. For a practical iteration problem, the key point is determining the corresponding nonlinear mapping relation.

In this paper, we propose a microwave-photonics iterative nonlinear gain (MING) model for analyzing envelope dynamics of OEOs. We demonstrate that the envelope amplitude evolution in OEOs can be transformed to iterated processes based on a determined nonlinear mapping relation called open-loop input to output amplitude mapping (IOAM) relation. Results show that degree of OEO nonlinearity is connected to the slope value of IOAM at a special fixed-point. OEO behaviors like a linear structure if the slope value is relatively large, linear features of homogeneity, monotonicity and stability are all satisfied. However, these linear features are gradually lost with the decrease of the slope value. Specifically, OEO is unstable when the slope value is less than a general threshold value of $- 1$. Behaviors of OEO with different slope values are theoretically analyzed and confirmed by both simulations and experiments. Furthermore, experiments show that the proposed model is suitable to the OEO with complex nonlinear properties. In addition, bifurcation phenomena caused by injected microwave signals are evidenced.

2. Theory

Figure 1(a) shows the structure under study, which is a single loop OEO with a switched point S and an injected port. The switched point S allows OEO to be switched between open-loop mode and closed-loop mode, and an external microwave signal can be injected into the OEO loop through the input/injected port. To make the analysis clear, the external microwave signal is named as the input signal and the injected signal corresponding to the open-loop and closed-loop respectively. The employed Mach-Zehnder modulator (MZM) is biased at the positive quadrature and a filter is used to remove the harmonic components.

 

Fig. 1. Diagram of the proposed theory. (a) shows the structure of the OEO under study. (b) shows the introduced graphical technique for resolving the iterated equation. The red curve and black line are IOAM and $\textrm{y} = V - {V_{in}}$ respectively, and the black point represents the fixed point. The trajectories (blue arrow line) show the oscillating processes. (c) shows that the linear features of homogeneity, monotonicity and stability are gradually lost with the decrease of the slope of IOAM at fixed point.

Download Full Size | PDF

When point S is disconnected, OEO is in open-loop mode which is a simple two port optoelectronic link, and the amplitude nonlinear property is clear. Assuming the total time delay corresponding to the physical length of the loop is $\tau $, and the frequency-dependent phase caused by the dispersive component is $\phi (\omega )$. Considering that an input signal with amplitude V is constantly input into the loop, and assuming that input frequency ${\omega _0}$ meets the oscillation condition of ${\omega _0}\tau + \phi ({\omega _0}) = 2m\pi ,$ $m = 1,2,3,\ldots $[1], the output amplitude is a function of V and can be given by [1,5,34]:

Next, we discuss the closed-loop OEO by connecting the switch point S while an injected signal with amplitude of ${V_{in}}$ and frequency of ${\omega _0}$ is constantly injected into the loop. Differ from the open-loop, the injected signal is circulated in the closed-loop, thus the output amplitude envelope is no longer a constant and is separated by the interval of total time delay of OEO $\tau$, which can be given by:

It is obvious that Fig. 1(b) indicates a case of a stable OEO, and the solution sequence $\{{{V_n}} \},n = 0,1,2,\ldots $ converges to the fixed point ${V^\ast }$. However, for different OEO loops, the IOAM and injected amplitude can be different which brings a different iterated diagram compared to Fig. 1(b) and causes the different nonlinearity of the loop. In general, the type of iterated diagram and the degree of nonlinearity is determined by the relative slope value of IOAM at the fixed point ${F^{\prime}_{open}}({V^\ast })$. Here, three linear features of homogeneity, monotonicity and stability are introduced to help us to analyze the nonlinear feature of the OEO. Homogeneity means that the obtained gain after oscillating single cycle in the loop for any oscillation microwave signal is identical, monotonicity means that the oscillation amplitude monotonically increases during the oscillating process, and stability indicates that the oscillation amplitude is converging [11]. As shown in Fig. 1(c), OEO behaves like a linear structure when ${F^{\prime}_{open}}({V^\ast })$ is relatively large, then nonlinear features emerge and become increasingly significant with the deceasing of ${F^{\prime}_{open}}({V^\ast })$, the linear features of homogeneity, monotonicity and stability are gradually lost.

To discuss the relationship between the slope value ${F^{\prime}_{open}}({V^\ast })$ and iterated type, some simulations are implemented in the next section.

3. Simulations

To simplify simulations, assuming that EA module works at linear operating range (${G_A}{f_{EA}}(V) = {G_A}V$). Considering normalized injected amplitude ${x_{in}} = {G_A}{V_{in}}{\pi / {{V_\pi }}}$, normalized oscillation amplitude ${x_{n - 1}} = {G_A}{V_{n - 1}}{\pi / {{V_\pi }}}$ and ${x_n} = {G_A}{V_n}{\pi / {{V_\pi }}}$, Eq. (3) can be further simplified into a normalized form:

3.1 Linear type of stable state

Linear features dominate OEO loop if $F^{\prime}({x^\ast })$ is relatively large. As shown in Fig. 1(c), type of OEOs with ${F^\prime }({x^\ast }) \approx {G_s}$ is referred as the linear type of stable state. Figure 2(a) shows a simulated iterated map based on Eq. (4) with ${G_s} = 0.7$ and ${x_{in}} = 0.1$. The red curve and black line represent the normalized IOAM $F(x)$ and $y = x - {x_{in}}$, respectively. As ${G_s} < 1$ and ${x_{in}}$ is relatively small, the fixed point ${x^\ast }$ locates in the linear area of IOAM which means ${F^\prime }({x^\ast }) \approx F^{\prime}(0) = {G_s}$, thus IOAM can be considered as a line $F(x) = {G_s}x$ in interval $(0,{x^\ast })$. Blue arrows are trajectories which indicate the oscillation processes. Let ${x_{m - 1}} = {x^\ast } - \Delta x$ and ${x_m} = {x^\ast } - k\Delta x$, $0 < k = {G_s} < 1$ is obtained. Thus, OEO shows strong linear features as follow (Detailed deviation see in Appendix 6.1.1):

 

Fig. 2. Linear type of stable state. (a) shows the iterated map, the red curve represents normalized IOAM and black line represents $y = x - {x_{in}}$, the blue arrows show the trajectories. (b) shows the simulated microwave oscillating signal (blue curve) and the simulated closed-loop gain (red curve).

Download Full Size | PDF

3.2 Quasi-linear type of stable state

Nonlinearity shows up if $F^{\prime}({x^\ast })$ decreases. As shown in Fig. 1(c), type of OEOs with $0 < F^{\prime}({x^\ast }) < {G_s}$ is referred as the quasi-linear type of stable state. Figure 3(a) shows a simulated iterated map with ${G_s} = 1.5$ and ${x_{in}} = 0.001$. The fixed point (black point) ${x^\ast }$ is located in the positive nonlinear area of IOAM $F(x)$ (Red curve), and $F(x)$ can be considered as a concave curve ($F^{\prime\prime}(x) < 0$ if $x \in (0,3.51)$). Trajectories (blue arrows) demonstrate the oscillation processes. Let ${x_{m - 1}} = {x^\ast } - \Delta x$ and ${x_m} = {x^\ast } - k\Delta x$, the convergence constant k meets the inequation $0 < F^{\prime}({x^\ast }) < k \le {{F({x^\ast })} / {{x^\ast }}} \le 1$. Thus, we have (Detailed deviation see in Appendix 6.1.2):

 

Fig. 3. Quasi-linear type of stable state. (a) shows the iterated map, the red curve represents normalized IOAM and black line represents $y = x - {x_{in}}$, the blue arrows show the trajectories. (b) shows the simulated microwave oscillating signal (blue curve) and the simulated closed-loop gain (red curve).

Download Full Size | PDF

The blue curve in Fig. 3(b) shows the corresponding simulated oscillation signal, which is actually the class OEOs described by Yao with quasi-linear model [1], noise in OEO had been considered as a tiny injected signal before EA [1,8,14], and the oscillation amplitude sequence ${\{{{x_n}} \}_{n = 0,1,2,..}}$ monotonically converged to the fixed point. The closed-loop gain (shown in red curve of Fig. 3(b)) monotonically decreased from ${G_s}$ to the unity due to the nonlinear effects of OEO loop.

3.3 Middle state

Nonlinearity increased if $F^{\prime}({x^\ast })$ further decreases. As shown in Fig. 1(c), when $- 1 < {F^\prime }({x^\ast }) < 0$, OEO is under a specific state which is named as the middle state (between stable and unstable states) in this paper. On the one hand, the middle state conserves the feature of stable state, the oscillation amplitude is eventually convergent. On the other hand, the middle state already has the feature of unstable state, oscillation amplitude non-monotonically oscillates before converging. Figure 4(a) shows a simulated iterated map case of the middle state with ${G_s} = 2.2$ and ${x_{in}} = 0.05$. The fixed point (black point) ${x^\ast }$ locates in the negative nonlinear area of IOAM $F(x)$ (red curve), and the slope value ${F^\prime }({x^\ast })$ is ∼$- 0.89$.

 

Fig. 4. Middle state. (a) shows the iterated map, the red curve represents normalized IOAM and black line represents $y = x - {x_{in}}$, the blue arrows show the trajectories. (c) shows the simulated microwave oscillating signal (blue curve) and the closed-loop gain (red curve).

Download Full Size | PDF

The trajectories (blue arrows) describe the oscillating processes, the oscillation amplitude firstly increases on the left side of ${x^\ast }$, and then flips into the right side of ${x^\ast }$ due to the negative slope area. Let ${x_{m - 2}} = {x^\ast } - \Delta x$, ${x_{m - 1}} = {x^\ast } - {k_1}\Delta x$, and ${x_m} = {x^\ast } - {k_2}\Delta x$. Considering ${x_{m - 2}}$ satisfied $F({x_{m - 2}}) > {x^\ast } - {x_{in}}$, we have ${k_1} < 0$. In addition, around the nearby of ${x^\ast }$, we have $0 < {k_2} = {|{F^{\prime}({x^\ast })} |^2} < 1$. Thus, we have (Detailed deviation seen in Appendix 6.1.3):

The blue curve in Fig. 4(b) shows the corresponding simulated oscillation signal whose envelope amplitude oscillates around ${x^\ast }$ before converging. The red curve in Fig. 4(b) shows the closed-loop gain ${G_k}$ which is not a monotonic but an oscillating sequence around $[F({x^\ast }) - {x_{in}}]/{x^\ast }$. Note that, ${G_k} < 1$ exists in some oscillating cycles, which is another sign that implies a strong nonlinearity.

For OEOs under middle state, oscillation amplitude sequence $\{{{x_n}} \}$ are separated into the even sequence $\{{{x_{2n}}} \}$ and odd sequence $\{{{x_{2n - 1}}} \}$ which are on the two sides of ${x^\ast }$ respectively. Both $\{{{x_{2n}}} \}$ and $\{{{x_{2n - 1}}} \}$ converge to ${x^\ast }$ due to $|{{F^\prime }({x^\ast })} |< 1$.

Furthermore, for the linear type, quasi-linear type of stable state and middle state, the building-up time of OEO is also determined by $|{F^{\prime}({x^\ast })} |$. Assuming that ${x_{m - 1}}$ is near the fixed-point and let ${x_{m - 1}} = {x^\ast } - \Delta x$, the oscillation amplitude after n OEO cycles can be given by (Detailed deviation see in Appendix 6.1.3):

3.4 Bistable state

When the slope value at fixed point ${F^\prime }({x^\ast })$ further decreases and crosses the threshold value of $- 1$, the nonlinear feature will affect the final OEO output and lead to an unstable state such as bistable state. Under the bistable state, even sequence $\{{{x_{2n}}} \}$ and odd sequence $\{{{x_{2n - 1}}} \}$ converge to two inequal values respectively. Figure 5(a) shows an improved iterated map with ${G_s} = 2.65$ and ${x_{in}} = 0.001$ based on this separation. The red curve represents $\textrm{y} = F(x) + {x_{in}}$, black curve is the symmetrical curve of red curve about line $y = x$. There are 3 intersections between red curve and black curve. One of them is the origin fixed point $x_{}^\ast $(black point in Fig. 5(a)). The other two intersections $x_{2 - 1}^\ast $ and $x_{2 - 2}^\ast $ (red points) are named as 2^nd order fixed points, and they meet relations of $x_{2 - 1,2}^\ast{=} F(x_{2 - 2,1}^\ast ) + {x_{in}}$ and $x_{2 - 1,2}^\ast{=} F[F(x_{2 - 1,2}^\ast ) + {x_{in}}] + {x_{in}}$. Blue arrows and purple arrows are trajectories showing that odd sequence $\{{{x_{2m - 1}}} \}$ and even sequence $\{{{x_{2m}}} \}$ approach to the nearby of $x_{2 - 1}^\ast $ and $x_{2 - 2}^\ast $ respectively. Figure 5(b) shows the zoom-in view of Fig. 5(a), where $\{{{x_{2m - 1}}} \}$ oscillates around $x_{2 - 1}^\ast $. Let ${x_{2m - 5}} = x_{2 - 1}^\ast{-} \Delta x$ and ${x_{2m - 1}} = x_{2 - 1}^\ast{-} {k_4}\Delta x$, ${k_4} = {|{F^{\prime}(x_{2 - 1}^\ast )F^{\prime}(x_{2 - 2}^\ast )} |^2}$ is obtained near $x_{2 - 1}^\ast $, thus threshold condition $|{F^{\prime}(x_{2 - 1}^\ast )F^{\prime}(x_{2 - 2}^\ast )} |= 1$ is obtained. OEO is in the bistable state provided that ${F^\prime }({x^\ast }) < - 1$ and $|{F^{\prime}(x_{2 - 1}^\ast )F^{\prime}(x_{2 - 2}^\ast )} |< 1$, and we have (Detailed deviation seen in Appendix 6.1.4):

 

Fig. 5. Bistable state. (a) shows an improved method for analyzing bistable state. The red curve represents $\textrm{y} = F(x) + {x_{in}}$, the black curve is the symmetrical curve of red curve about line $y = x$. The black point is the origin fixed-point ${x^\ast }$, and red points are the 2^nd order fixed points $x_{2 - 1}^\ast $ and $x_{2 - 2}^\ast $. Blue arrows and green arrows are trajectories corresponding to odd amplitude sequence $\{{{x_{2n - 1}}} \}$ and even amplitude sequency $\{{{x_{2n}}} \}$ respectively. (b) is the zoomed-in view of the (a) and indicates that the $\{{{x_{2n - 1}}} \}$ converges to $x_{2 - 1}^\ast $. (c) shows the simulated microwave oscillation signal which is separated into the even cycles (green curve) and odd cycles (blue curve), the closed-loop gain (red curve) oscillates constantly.

Download Full Size | PDF

Green and blue curves show in Fig. 5(c) represent the simulated oscillation microwave signal corresponding to even and odd oscillating cycles respective, which demonstrates that the final envelope amplitudes of the adjacent OEO cycles are two different values of $x_{2 - 1}^\ast $ and $x_{2 - 2}^\ast $. Closed-loop gain ${G_k}$ (red curve) constantly oscillates, which indicates the significant instability.

When the ${F^\prime }({x^\ast })$ further decreases and leads to $|{F^{\prime}(x_{2 - 1}^\ast )F^{\prime}(x_{2 - 2}^\ast )} |> 1$, amplitude sequence of OEO will further bifurcate, and bring a further unstable state such as 4-points cycle, 8-points cycle etc., even chaos.

Note that, the building-up time of bistable state OEO is also determined by $|{F^{\prime}(x_{2 - 2}^\ast )F^{\prime}(x_{2 - 1}^\ast )} |$. Assuming that ${x_{2m - 1}}$ is near $x_{2 - 1}^\ast $ and ${x_{2m - 2}}$ is near $x_{2 - 2}^\ast $. Let ${x_{2m - 1}} = x_{2 - 1}^\ast{-} {C_1}\Delta x$ and ${x_{2m}} = x_{2 - 2}^\ast{-} {C_2}\Delta x$, the oscillation amplitude after 2n OEO cycles can be given by (Detailed deviation see in Appendix 6.1.4):

3.5 Bifurcation diagram

Above simulations in subsections 3.1-3.4 show that degree of OEO nonlinearity increased with the decrease of ${F^\prime }({x^\ast })$. Thus, the state of OEO can be changed by adjusting IOAM and injected amplitude ${x_{in}}$ because ${F^\prime }({x^\ast })$ is determined by them. Specifically, IOAM is only the function of open-loop small signal ${G_s}$ if OEO only concerns the MZM nonlinear effects based on Eq. (4). Under this condition, the bifurcation phenomenon can be observed from two dimensions of ${G_s}$ and ${x_{in}}$.

First, we discuss the bifurcation behavior of OEOs with different small-signal gains ${G_\textrm{s}}$. Figure 6(a) shows simulated bifurcation diagram of a specifically case with ${x_{in}} = 0$, OEO state varies from stable to bistable, 4-points cycle states etc. even chaos with the increase of ${G_\textrm{s}}$. The threshold gain values of ${G_\textrm{s}}$ between stable and bistable states, bistable and 4-points cycle sates are 2.31 and 2.7 respectively, because $|{F^{\prime}({x^\ast })} |= 1$ when ${G_\textrm{s}} \approx 2.31$ and $|{F^{\prime}(x_{2 - 1}^\ast )F^{\prime}(x_{2 - 2}^\ast )} |= 1$ when ${G_\textrm{s}} \approx 2.7$.

 

Fig. 6. Bifurcation behavior of OEO. (a) Bifurcation diagram with changed open-loop small-signal gain ${G_s}$ and without injected signal. (b) Red curve and blue curve represent IOAM and slope value of IOAM with ${G_\textrm{s}} = 2$ respectively. Red points are the fixed points and ordinates of the blue points are the slope value of IOAM at the corresponding fixed points. (c) Bifurcation diagram $({G_s} = 2)$ with changed injected amplitude. (d) Bifurcation diagram $({G_s} = 2.1)$ with changed injected amplitude.

Download Full Size | PDF

However, the threshold gain values of 2.31 and 2.7, which fits well with the results obtained by Chembo [12,13], are invalid when face to the OEOs which concern the nonlinear effects in additional to MZM. In fact, it will be evidenced that the threshold gain value is significantly increased due to the saturation characteristic of the EA module in Experiments 4.1. This can be easy to understand from the view of the proposed MING model, because the general threshold condition of $|{F^{\prime}({x^\ast })} |= 1$ is determined by IOAM $F(x)$, and IOAM is shaped by the nonlinear effects of EA module.

Then, we discuss the bifurcation behavior of OEOs with different injected amplitudes. Red curve and blue curve in Fig. 6(b) show simulated $F(x)$ and $F^{\prime}(x)$ with ${G_\textrm{s}} = 2$ respectively. The red points show the fixed points corresponding to the ${x_{in}} = 0$ and ${x_{in}} = 1.5$ respectively, and the ordinates of blue points are the corresponding $F^{\prime}({x^\ast })$. It is obviously that $|{F^{\prime}({x^\ast })} |< 1$ when ${x_{in}} = 0$, which indicates a stable state. Then, $x_{}^\ast $ moves along the positive direction of the horizontal axis with the increase of ${x_{in}}$, OEO is unstable when ${x^\ast }$ reach to the area with $F^{\prime}(x) < - 1$. Noticed that, only part of area meets $F^{\prime}(x) < - 1$, which means that OEO state can be stable again with the further increase of ${x_{in}}$. This phenomenon indicates that the injected amplitude is another variety of the OEO stability in addition to small signal gain ${G_\textrm{s}}$, because the OEO state cannot be stable again with the increase of the ${G_\textrm{s}}$ as shown in Fig. 6(a).

Figure 6(c) shows the corresponding bifurcation diagram with ${G_\textrm{s}} = 2$. As expected, the stable state becomes unstable then back into stable state again with the increase of ${x_{in}}$. Figure 6(d) shows another simulated bifurcation diagram with ${G_\textrm{s}} = 2.1$ and changed ${x_{in}}$, due to the relatively large ${G_\textrm{s}}$, condition of $F^{\prime}(x_{2 - 1}^\ast )F^{\prime}(x_{2 - 2}^\ast ) < - 1$ is satisfied in some area which leads to 4-points cycle state.

Furthermore, in the area between two spotted red line as shown in Figs. 6(c-d), there are both positive and negative oscillation amplitudes, which implies a phase shift of $\pi $ between the adjacent cycles of OEO.

4. Experiments

The analytical and numerical analysis are confirmed by the following experimental results. Subsection 4.1 shows that the general threshold condition between the stable and unstable states is the slope value equals to $- 1$. Subsection 4.2 shows nonlinearity of OEO increases with the decrease of the slope value, and subsection 4.3 confirms the bifurcation diagrams shown in Fig. 6(c-d).

4.1 General threshold condition between stable and unstable states

The experimental results shown in Fig. 7 confirm that OEO is stable when the slope value of IOAM at fixed point ${F_{open}}({V^\ast })$ is larger than threshold value of $- 1$, and the final oscillation amplitude converges into the fixed point.

 

Fig. 7. Amplitude value of stable OEO with different small-signal gain. Colorful curves in (a-b) show measured IOAMs with difference small-signal gain, which are shaped by nonlinear effects of both the MZM and the EA module. Red points show the corresponding fixed points, and the normalized abscissas of the fixed points are extracted and are employed as the calculating results (red points) in (c). Slope values at fixed points shown in (a-b) are all larger than $- 1$, thus all corresponding oscillating signals are stable. After employing FFT, the amplitude measurements are obtained and shown in the blue curve of (c), which meets the calculating results (red points) well.

Download Full Size | PDF

The employed set up is shown in Fig. 1(a), an MZM with half-wave voltage of about 2.5 V is biased at positive quadrature, bandwidth of the filter is 10MHz@7 GHz, total time delay is about $0.3\mu s$. Two EAs with relatively small linear operating range (from $- 65$ to $18$ dBm) and a tunable attenuator (ATT, LDA-5018 V of Vaunix) are employed to form the tunable EA module. Thus, the IOAM of OEO loop can be changed by adjusting the gain of the ATT. The main experimental steps are as follow:

We performed several measurements with different small-signal gains from 8.04 to 15.55 dB. The colorful curves in Figs. 7(a) and 7(b) show the corresponding IOAM measurements which are shaped by both MZM and EA module nonlinear effects, black spotted lines are $y = x$, the intersections (red points) between IOAM and $y = x$ are the fixed points. Slope values at fixed points are all larger than $- 1$, thus the oscillation signals are all stable.

The measured normalized amplitude values of oscillation signal are obtained by employing Fourier transform FFT to the OSC recorded signals and are shown in the blue curve of Fig. 7(c). The agreement of these experiments with the corresponding fixed points (red points) is good, which confirms that the oscillation amplitude of the stable OEO converges into the fixed point.

Then, we further increased the small-signal gain of OEO loop from 16.4 to 19.9 dB. Colorful curves and fixed points in Fig. 8(a) show the corresponding IOAM measurements and fixed points. Green spotted lines with slope of $- 1$ are illustrated to provide the comparison between the slope of IOAM at fixed points and slope of $- 1$. Figures 8(b)–8(d) show the corresponding OSC recorded oscillation signals. When the slope value of IOAM at fixed point is larger than $- 1$ as shown by fixed point (b), oscillation signal is stable. When slope values of IOAM at fixed point are smaller than $- 1$ as shown by fixed points (c-d), oscillation signals are unstable.

 

Fig. 8. Threshold condition between stable and unstable OEOs. Colorful curves in (a) show IOAMs with different small-signal gain, and red points show the corresponding fixed points. Slopes of the green spotted lines are all $- 1$. (b-d) show the oscillation signals corresponding to the different $F^{\prime}({x^\ast })$. (b) illustrates that OEO is under stable state when slope value of IOAM at fixed point is bigger than $- 1$. (c-d) illustrate that OEO is unstable when the slope value is smaller than $- 1$.

Download Full Size | PDF

As mentioned before, for the class of OEOs which only concern MZM nonlinear effects and without injected signals, the threshold voltage gain between stable and unstable states is 2.31(∼7.23 dB, which is also confirmed by the experiments shown in Appendix 6.2). And this threshold value is invalid in the above experiments due to the nonlinear effects introduced by EA module. Figure 8(a) shows that OEO maintains a stable state with small-signal gain of 16.44 dB. However, the general threshold slope value of $- 1$ is still satisfied.

4.3 Oscillating behaviors with the different slope value

The second experiment is designed to investigate the oscillating behaviors of OEOs with different slope values. The major setup of this experiment is the same as subsection 4.1, while two EAs in the EA module are replaced by EAs with large operating range (from $- 65$ to $23$ dBm), so that saturation of EAs is not concerned, and the negative slope area of IOAM can be generated with a relatively small gain. Furthermore, to realize the repetitive measurements of the oscillating process, a switched laser module is employed. As shown in Fig. 9(a), an additional MZM with half-wave voltage of 7 V is employed after the laser, the driven signal is a square wave with repetition frequency of 5000 Hz, rise/fall edge of 2 ns, duty ratios of 90% (81150A of Agilent Technologies). This additional MZM is biased at negative quadrature, and the output optical power varies with the driven voltage.

 

Fig. 9. Different oscillating types corresponding to different $F^{\prime}({x^\ast })$.(a) shows the structure of the employed switched laser module. (b) Blue curve shows the IOAM, the fixed point moves along the positive direction of horizontal axis with the increasing of injected amplitude, corresponding amplitudes of the oscillation signals are obtained by employ FFT and are shown as the red points. (c-e) Real-time signals (up) and STFT results (bottom) show the oscillating processing corresponding to the different $F^{\prime}({x^\ast })$. (c) shows a case of quasi-linear type and (d-e) show two cases of middle state.

Download Full Size | PDF

First, we fixed the driven voltage of MZM as shown in Fig. 9(a) at 0 V, set OEO under open-loop and adjusted small signal gain to 0.2 dB. The IOAM is obtained from VNA and is shown in blue curve of Fig. 9(b).

After measuring the IOAM, we set the OEO under closed-loop and opened the square driven signal of the additional MZM as shown in Fig. 9(a). Then, a microwave signal with the frequency equal to the oscillation signal of OEO (∼7 GHz), which is generated by a tunable microwave signal source (SMW200A of Rohde&Schwarz), was injected into the closed-loop OEO through the input port as shown in Fig. 1(a). The oscillating process of OEO was obtained and recorded by an OSC with a sampling rate of 25 GSa/s.

We performed several measurements with different injected amplitudes (from ∼30 to ∼220 mV), thus the fixed point moves correspondingly, and the slope value at fixed point decreases with increase of the injected amplitude. However, as the small-signal gain is relatively small, the slope values at different fixed points are all larger than $- 1$, thus all oscillation signals are eventually stable. Red points in Fig. 9(b) show the stable amplitude measurements, which are obtained from the short time Fourier transform (STFT) results of the oscillation signal. Red point meets the IOAM well and indicates that the oscillation amplitude converges to the fixed point.

Furthermore, three cases of the oscillating processes corresponding to points (c-e) shown in Fig. 9(b) are provided in up figures of Fig. 9(c-e), and bottom figures show the corresponding STFT results. The slope value at fixed point (c) is positive, which leads to a quasi-linear type oscillating process. As shown in Fig. 9(c), oscillation amplitude monotonically increases and converges into the fixed point, which fits the simulation shown in Fig. 3 well and implies a quasi-linear OEO type. The slope value at fixed point (d) is negative, thus the OEO is under middle state. As shown in Fig. 9(d), oscillation amplitude oscillates around the fixed point before converging, which fits the simulation shown in Fig. 4 well. The slope value at fixed point (e) is closer to $- 1$ compare with point (c), thus the oscillating process shown in Fig. 9(e) needs more time before converging. Experiments shown in Fig. 9 confirm that the degree of OEO nonlinearity increases with the decrease of the slope value at fixed point.

4.4 Bifurcation phenomenon caused by injected signal

Figures 10 and 11 show the experimental bifurcation diagrams corresponding to the simulations shown in Figs. 6(c-d). The major setup of this experiment is the same as subsection 4.1, while two EAs in the EA module are replaced by EAs with large operating range (from $- 65$ to $23$ dBm).

 

Fig. 10. (a) Experimental bifurcation diagram with small signal gain ${G_s}$ of 2 and with different injected amplitudes. (b-f) show the oscillation signals (up) and the corresponding STFT results (bottom).

Download Full Size | PDF

 

Fig. 11. (a) Experimental bifurcation diagram with small signal gain ${G_s}$ of 2.1 and with different injected amplitudes. (b-f) show the oscillation signals (up) and the corresponding STFT results (bottom).

Download Full Size | PDF

First, the open-loop small-signal voltage gain ${G_s}$ was set at ∼2. A microwave signal with the frequency equal to the oscillation signal (∼7 GHz) was injected into the closed-loop OEO, the corresponding oscillation signals were collected by an OSC with a sampling rate of 25 GSa /s. We did several measurements with different injected amplitude. After calculating STFT of the collected signals, the bifurcation diagram can be obtained and shown in Fig. 10(a). The experimental results agree well with the simulations as shown in Fig. 6(c), which confirms that the stability of OEO is a function of the injected amplitude.

Furthermore, oscillation signals (up) and STFT results (bottom) corresponding to different injected amplitudes are shown in Figs. 10(b-f) to illustrate the details. Figures 10(b) and 10(f) show two stable output results, and Figs. 10(c-e) show three bistable results. Note that there is a significant gap between adjacent OEO cycles as shown in Figs. 10(d-e), which indicates that there is a phase shift of $\pi $ between the adjacent cycles of the OEO as expected.

Then the small-signal voltage gain ${G_s}$ is adjusted to ∼2.1, after the same experimental processes above, the bifurcation diagram is obtained and is shown in Fig. 11(a). Experimental results fit well with the simulation shown in Fig. 6(d). Furthermore, oscillation signals (up) and STFT results (bottom) corresponding to different injected amplitudes are shown in Figs. 11(b-f) to illustrate the details. Figure 11(f) shows a stable state, Figs. 11(b) and 11(e) show two bistable states and Figs. 11(c-d) show two 4-points cycle states. The experimental results in Figs. 10–11 show that the injected amplitude is a variable of OEO stability as well as the small signal gain.

5. Conclusion

In this paper, we have investigated the nonlinear envelope dynamics of OEOs via the proposed MING model. We have connected oscillating processes of OEO with the trajectories of geometrical iterated map based on IOAM. We have demonstrated that stability of OEO is determined by slope value of IOAM at fixed point. The dynamic behavior of OEO with different slope value are classified discussed in the Simulation section, results show that the degree of OEO nonlinearity increases with the decreasing of the slope value of IOAM at fixed point: the linear features of homogeneity, monotonicity and stability are generally lost.

The experimental results agree well with the theory and simulations. First, because the IOAM has already considered the total nonlinear effects of the OEO loop, the proposed model can be applied to a practical OEO with complex nonlinear effects. OEO which is affected by the saturation characteristic of EA module has been tested under different small signal gains. Results show that the oscillation amplitude of stable OEO converges into the fixed point and confirm that the general threshold condition between stable and unstable state is the slope value equal to $- 1$. Secondly, the oscillating processes of OEO with the different slope values have been investigated, results clearly evidenced that the degree of OEO nonlinearity increased with the decreasing of the slope value. Lastly, bifurcation phenomenon caused by injected microwave signals have been confirmed, which fits the simulation well and shows that the injected amplitude is a variable of OEO stability in addition to the small-signal gain.

6. Appendix

6.1 Theoretical derivation

In this subsection, homogeneity, monotonicity and stability of OEO with different ${F^\prime }({x^\ast })$ are detailed discussed.

6.1.1 Linear type

OEO with ${F^\prime }({x^\ast }) \approx F^{\prime}(0) = {G_s} < 1$ belongs to the linear type and behaviors like a linear structure: homogeneity and monotonicity are satisfied during the oscillating process, and final output amplitude is stable.

First, we discuss the homogeneity, as shown in Fig. 2(a), when ${x^\ast }$ locates at the linear area of IOAM, ${F^\prime }({x^\ast }) \approx F^{\prime}(0) = {G_s} < 1$ is obtained, due to the smoothness of IOAM, we have that $F(x) = {G_s}x$ in the interval $(0,{x^\ast })$, and the iterated equation can be given by,

(8)$${x_n} = {G_s}{x_{n - 1}} + {x_{in}}.$$

Thus, the closed-loop gain for any oscillation signal ${G_k} = {{F({x_k})} / {{x_k}}} = {G_s}$ is a constant which indicates the homogeneity of OEO loop.

Second, we discuss the monotonicity, considering two adjacent oscillating amplitude ${x_{m - 1}} = {x^\ast } - \Delta x$ and ${x_m} = {x^\ast } - k\Delta x$, combing the definition of fixed point ${x^\ast } = {G_s}{x^\ast } + {x_{in}}$, we have that

(9)$$k = \frac{{{x^\ast } - {x_m}}}{{{x^\ast } - {x_{m - 1}}}} = \frac{{{G_s}{x^\ast } + {x_{in}} - ({G_s}{x_{m - 1}} + {x_{in}})}}{{{x^\ast } - {x_{m - 1}}}} = {G_s}.$$

Thus, the convergence constant k is unchanged during the oscillating process with $0 < k = {G_s} < 1$. Therefore, ${x_m} > {x_{m - 1}}$ is always satisfied which confirms the monotonicity.

Third, $0 < k = {G_s} < 1$ also implies that $\mathop {\lim }\limits_{n \to \infty } {x_n} = {x^\ast }$ which confirms that the OEO is stable.

6.1.2 Quasi-linear type

OEO with $0 < {F^\prime }({x^\ast }) < {G_s}$ belongs to the quasi-linear type: nonlinear feature emerges in OEO: homogeneity is not satisfied, but monotonicity and stability are still satisfied.

First, we discuss the monotonicity and the stability, as shown in Fig. 3(a), considering two adjacent oscillating amplitude ${x_{m - 1}} = {x^\ast } - \Delta x$ and ${x_m} = {x^\ast } - k\Delta x$, we have that

(10)$$\begin{aligned} k = \frac{{{x^\ast } - {x_m}}}{{{x^\ast } - {x_{m - 1}}}} &= \frac{{F({x^\ast }) - {x_{in}} - [F({x_{m - 1}}) - {x_{in}}]}}{{{x^\ast } - {x_{m - 1}}}}\\ &= \frac{{F({x^\ast }) - F({x_{m - 1}})}}{{{x^\ast } - {x_{m - 1}}}}. \end{aligned}$$

Because ${F^\prime }({x^\ast }) < 0$ is satisfied in interval $(0,3.51)$, IOAM is a smooth concave curve. Thus,

(11)$$F({x^\ast }) - F^{\prime}({x^\ast })({x^\ast } - x) > F(x)$$

is satisfied in interval $(0,{x^\ast })$. Let $x = {x_{m - 1}}$, substitute Eq. (10) and condition $0 < {F^\prime }({x^\ast })$ in Eq. (11), we have

(12)$$0 < F^{\prime}({x^\ast }) < k. $$

Due to IOAM is a smooth concave curve again, we have that

(13)$$F(x) \ge \frac{{F({x^\ast })}}{{{x^\ast }}}x$$

is satisfied in interval $[0,{x^\ast }]$. Let $x = {x_{m - 1}}$, substitute Eq. (13) into Eq. (10), we have

(14)$$\begin{aligned} k = \frac{{F({x^\ast }) - F({x_{m - 1}}) }}{{{x^\ast } - {x_{m - 1}}}} &\le \frac{{F({x^\ast }) - \frac{{F({x^\ast }) }}{{{x^\ast }}}{x_{m - 1}}}}{{{x^\ast } - {x_{m - 1}}}}\\ &= \frac{{F({x^\ast }) }}{{{x^\ast }}} = \frac{{{x^\ast } - {x_{in}}}}{{{x^\ast }}} \le 1. \end{aligned}$$

Combing Eqs. (12) and (14), we have $0 < k \le 1$ which indicates that ${x_m} \ge {x_{m - 1}}$ is always satisfied and $\mathop {\lim }\limits_{n \to \infty } {x_n} = {x^\ast }$. Therefore, monotonicity and stability are satisfied.

However, due to closed-loop gain ${G_k} = {{F({x_k})} / {{x_k}}}$ is no more a constant but a function of the oscillation amplitude, the homogeneity is not satisfied.

6.1.3 Middle state

OEO with $- 1 < {F^\prime }({x^\ast }) < 0$ belongs to the middle state: OEO shows a strong nonlinear feature: neither homogeneity nor monotonicity is satisfied, but the final output state is still stable.

First, the homogeneity is not satisfied just like quasi-linear type, ${G_k} = {{F({x_k})} / {{x_k}}}$ is a function of the oscillation amplitude.

Second, we discuss the monotonicity, as shown in Fig. 4(a), considering two adjacent oscillating amplitude ${x_{m - 2}} = {x^\ast } - \Delta x$ and ${x_{m - 1}} = {x^\ast } - {k_1}\Delta x$, we have that,

(15)$${k_1} = \frac{{F({x^\ast }) - F({x_{m - 2}}) }}{{{x^\ast } - {x_{m - 2}}}}.$$

At the begin of the oscillating processes, amplitude sequence is on the left side of fixed point, assuming $F({x_{m - 2}}) \le F({x^\ast })$, ${k_1} > 0$ is satisfied. Thus, amplitude sequence monotonically increases, then when amplitude reach to the area of $F(x) > F({x^\ast })$, amplitude will flip into the negative slope area of the IOAM. In the negative area, ${k_1} < 0$ is satisfied which indicates that the monotonicity is not satisfied, and amplitude sequence oscillates around the fixed point.

Third, we discuss the stability, considering amplitudes nearby ${x^\ast }$: ${x_{m - 2}} = {x^\ast } - \Delta x$, ${x_{m - 1}} = {x^\ast } - {k_1}\Delta x$ and ${x_m} = {x^\ast } - {k_2}\Delta x$, by Eq. (4) and ${x^\ast } = F({x^\ast }) + {x_{in}}$, we have that:

(16)$$\begin{aligned} {x_{m - 1}} &= F({x_{m - 2}}) + {x_{in}}\\ &= F({x^\ast } - \Delta x) + {x_{in}}\\ &= F({x^\ast }) - F^{\prime}({x^\ast })\Delta x + O(\Delta {x^2}) + {x_{in}}\\ &= {x^\ast } - F^{\prime}({x^\ast })\Delta x. \end{aligned}$$

Then, by the same process, we have

(17)$$\begin{aligned} {x_m} &= F({x_{m - 1}}) + {x_{in}}\\ &= F({x^\ast } - F^{\prime}({x^\ast })\Delta x) + {x_{in}}\\ &= {x^\ast } - {|{F^{\prime}({x^\ast })} |^2}\Delta x. \end{aligned}$$

Compare Eq. (17) with ${x_m} = {x^\ast } - k\Delta x$, we have $k = {|{F^{\prime}({x^\ast })} |^2}$, thus OEO is stable if $- 1 < {F^\prime }({x^\ast })$ and is unstable if ${F^\prime }({x^\ast }) < - 1$.

Furthermore, for the linear type, quasi-linear type of stable state and middle state, the building-up time of OEO is also determined by $|{F^{\prime}({x^\ast })} |$. Assuming that ${x_{m - 1}}$ is near the fixed-point and let ${x_{m - 1}} = {x^\ast } - \Delta x$, after combine Eqs. (9–10) and (16–17), the oscillation amplitude after n OEO cycles can be given by:

(18)$${x_{m + n - 1}} = {x^\ast } - {|{F^{\prime}({x^\ast })} |^n}\Delta x.$$

Equation (18) shows that more $|{F^{\prime}({x^\ast })} |$ close to threshold value of $1$, more oscillating cycles are needed before converging.

6.1.4 Bistable state

When slope value at fixed point meets relation of ${F^\prime }({x^\ast }) < - 1$, OEO is in the unstable state such as the bistable state. None of linear features of homogeneity, monotonicity and stability are satisfied.

Here we discuss the threshold condition of the bistable state. According the definition of 2^nd order fixed points, we have

(19)$$\begin{aligned} x_{2 - 2}^\ast &= F(x_{2 - 1}^\ast ) + {x_{in}},\\ x_{2 - 1}^\ast &= F(x_{2 - 2}^\ast ) + {x_{in}}. \end{aligned}$$

As shown in Fig. 5(b), let ${x_{2m - 5}} = x_{2 - 1}^\ast{-} \Delta x$ and ${x_{2m - 1}} = x_{2 - 1}^\ast{-} {k_4}\Delta x$. By Eq. (4), we have that:

(20)$$\begin{aligned} {x_{2m - 4}} &= F({x_{2m - 5}}) + {x_{in}}\\ &= F(x_{2 - 1}^\ast{-} \Delta x) + {x_{in}}\\ &= F(x_{2 - 1}^\ast ) - F^{\prime}(x_{2 - 1}^\ast )\Delta x + o(\Delta {x^2}) + {x_{in}}\\ &= x_{2 - 2}^\ast{-} F^{\prime}(x_{2 - 1}^\ast )\Delta x. \end{aligned}$$

Then, by the same process, we have that:

(21)$$\begin{aligned} {x_{2m - 3}} &= x_{2 - 1}^\ast{-} F^{\prime}(x_{2 - 2}^\ast )F^{\prime}(x_{2 - 1}^\ast )\Delta x,\\ {x_{2m - 2}} &= x_{2 - 2}^\ast{-} F^{\prime}(x_{2 - 2}^\ast ){|{F^{\prime}(x_{2 - 1}^\ast )} |^2}\Delta x,\\ {x_{2m - 1}} &= x_{2 - 1}^\ast{-} {|{F^{\prime}(x_{2 - 2}^\ast )F^{\prime}(x_{2 - 1}^\ast )} |^2}\Delta x. \end{aligned}$$

Compare Eq. (21) with ${x_{2m - 1}} = x_{2 - 1}^\ast{-} {k_4}\Delta x$, we have ${k_4} = {|{F^{\prime}(x_{2 - 2}^\ast )F^{\prime}(x_{2 - 1}^\ast )} |^2}$, thus unstable OEO with ${|{F^{\prime}(x_{2 - 2}^\ast )F^{\prime}(x_{2 - 1}^\ast )} |^2} < 1$ is in the bistable state, even amplitude sequence $\{{{x_{2m}}} \}$ and odd amplitude sequence $\{{{x_{2m - 1}}} \}$ converge into two values of $x_{2 - 2}^\ast $ and $x_{2 - 1}^\ast $ respectively.

Furthermore, the building-up time of the bistable OEO is also determined by $|{F^{\prime}(x_{2 - 2}^\ast )F^{\prime}(x_{2 - 1}^\ast )} |$. Assuming that ${x_{2m - 1}}$ is near $x_{2 - 1}^\ast $ and ${x_{2m - 2}}$ is near $x_{2 - 2}^\ast $. By Eq. (21), the oscillation amplitude after 2n OEO cycles can be given by:

(22)$$\begin{aligned} {x_{2m - 1 + 2n}} &= x_{2 - 1}^\ast{-} {C_1}{|{F^{\prime}(x_{2 - 1}^\ast )F^{\prime}(x_{2 - 2}^\ast )} |^n}\Delta x,\\ {x_{2m + 2n}} &= x_{2 - 2}^\ast{-} {C_2}{|{F^{\prime}(x_{2 - 1}^\ast )F^{\prime}(x_{2 - 2}^\ast )} |^n}\Delta x. \end{aligned}$$

Equation (22) shows that more $|{F^{\prime}(x_{2 - 1}^\ast )F^{\prime}(x_{2 - 2}^\ast )} |$ close to threshold value of $1$, more oscillating cycles are needed before reaching the bistable state.

6.2 Another case about the threshold condition between stable and unstable states

In this subsection, two EAs employed in subsection 4.1 are replaced by two EAs with relatively large linear operating range (from $- 65$ to $23$ dBm). Thus, the nonlinear effects caused by EA module can be ignored.

Colorful curves in Fig. 12(a) demonstrate IOAMs with different small-signal gain from 6.55 to 8.48 dB, which are similar to ${J_1}$ as expected, red points show the corresponding fixed points. Figure 12(b) shows the zoom-in view of Fig. 12(a), one could note that slope value at point(c) is larger than −1, and slope values at point 12(d) and 12(e) are smaller than −1. Figures 12(c)–12(e) shows the oscillation signals corresponding to points in Figs. 12(c)–12(e) as shown in Fig. 12(b), respectively.

 

Fig. 12. Threshold condition between stable and unstable OEOs (only consider the MZM nonlinear). (a) shows IOAMs with different small signal gain, which are closed to ${J_1}$. Red points show the corresponding fixed points. (b) is the zoom-in view of (a), green spotted lines in (b) provide comparison between slope of IOAM at fixed points $F^{\prime}({x^\ast })$ and slope of −1. (b) shows that $F^{\prime}({x^\ast })$ at point c is large than −1, $F^{\prime}({x^\ast })$ at points d and e are less than −1. (c-d) shows the real-time microwave signals of the OEO with different fixed points, and verity that condition of $F^{\prime}({x^\ast }) > - 1$ leads to the stable OEOs, and $F^{\prime}({x^\ast }) < - 1$ leads to the unstable OEOs.

Download Full Size | PDF

Oscillation signal shown Fig. 12(c) is stable due to that the slope value is larger than −1, and oscillation signals shown Figs. 12(d)–12(e) are unstable due to the slope value is small than −1 as expected. Furthermore, corresponding small-signal gain of oscillation signal shown in Fig. 12(d) is 7.55 dB which is close to the voltage gain value of 2.31(∼7.23 dB). Experimental results confirmed again that threshold condition between stable and unstable states of OEO is $F^{\prime}({x^\ast }) ={-} 1$.

Funding

National Key Research and Development Program of China (2018YFB2201901, 2018YFB2201902, 2018YFB2201903); National Natural Science Foundation of China (61925505); Beijing Municipal Natural Science Foundation (Z210005).

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. X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B 13(12), 34–35 (1996).

2. X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32(7), 1141–1149 (1996). [CrossRef]  

3. X. S. Yao and L. Maleki, “Opto-electronic oscillator and its applications,” in International Topical Meeting on Microwave Photonics. MWP'96 Technical Digest. Satellite Workshop (Cat. No. 96TH8153) (IEEE, 1996), pp. 265–268.

4. Y. K. Chembo, D. Brunner, M. Jacquot, and L. Larger, “Optoelectronic oscillators with time-delayed feedback,” Rev. Mod. Phys. 91(3), 035006 (2019). [CrossRef]  

5. X. Zou, X. Liu, W. Li, P. Li, W. Pan, L. Yan, and L. Shao, “Optoelectronic oscillators (OEOs) to sensing, measurement, and detection,” IEEE J. Quantum Electron. 52(1), 1–16 (2016). [CrossRef]  

6. L. Maleki, “The optoelectronic oscillator,” Nat. Photonics 5(12), 728–730 (2011). [CrossRef]  

7. D. Strekalov, D. Aveline, N. Yu, R. Thompson, A. B. Matsko, and L. Maleki, “Stabilizing an optoelectronic microwave oscillator with photonic filters,” J. Lightwave Technol. 21(12), 3052–3061 (2003). [CrossRef]  

8. Y. Ji, X. Yao, and L. Maleki, “Compact optoelectronic oscillator with ultra-low phase noise performance,” Electron. Lett. 35(18), 1554–1555 (1999). [CrossRef]  

9. T. Davidson, P. Goldgeier, G. Eisenstein, and M. Orenstein, “High spectral purity CW oscillation and pulse generation in optoelectronic microwave oscillator,” Electron. Lett. 35(15), 1260–1261 (1999). [CrossRef]  

10. H. S. Black, “Stabilized feedback amplifiers,” Bell Syst. Tech. J. 13(1), 1–18 (1934). [CrossRef]  

11. A. V. Oppenheim, A. S. Willsky, S. H. Nawab, and G. M. Hernández, Signals & systems (Pearson Educación, 1997).

12. Y. K. Chembo, L. Larger, and P. Colet, “Nonlinear Dynamics and Spectral Stability of Optoelectronic Microwave Oscillators,” IEEE J. Quantum Electron. 44(9), 858–866 (2008). [CrossRef]  

13. 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]  

14. Y. K. Chembo, K. Volyanskiy, L. Larger, E. Rubiola, and P. Colet, “Determination of Phase Noise Spectra in Optoelectronic Microwave Oscillators: A Langevin Approach,” IEEE J. Quantum Electron. 45(2), 178–186 (2009). [CrossRef]  

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

16. L. Larger, “Complexity in electro-optic delay dynamics: modelling, design and applications,” Philos. Trans. R. Soc., A 371(1999), 20120464 (2013). [CrossRef]  

17. S. E. Hosseini, A. Banai, and F. X. Kärtner, “Low-drift optoelectronic oscillator based on a phase modulator in a Sagnac loop,” IEEE Trans. Microwave Theory Tech. 65(7), 2617–2624 (2017). [CrossRef]  

18. Y. Bao, E. Banyas, and L. Illing, “Periodic and quasiperiodic dynamics of optoelectronic oscillators with narrow-band time-delayed feedback,” Phys. Rev. E 98(6), 062207 (2018). [CrossRef]  

19. K. E. Callan, L. Illing, Z. Gao, D. J. Gauthier, and E. Schöll, “Broadband chaos generated by an optoelectronic oscillator,” Phys. Rev. Lett. 104(11), 113901 (2010). [CrossRef]  

20. J.-P. Goedgebuer, P. Levy, L. Larger, C.-C. Chen, and W. T. Rhodes, “Optical communication with synchronized hyperchaos generated electrooptically,” IEEE J. Quantum Electron. 38(9), 1178–1183 (2002). [CrossRef]  

21. A. Banerjee, L. A. D. de Britto, and G. M. Pacheco, “Analysis of injection locking and pulling in single-loop optoelectronic oscillator,” IEEE Trans. Microwave Theory Tech. 67(5), 2087–2094 (2019). [CrossRef]  

22. L. Bogataj, M. Vidmar, and B. Batagelj, “A feedback control loop for frequency stabilization in an opto-electronic oscillator,” J. Lightwave Technol. 32(20), 3690–3694 (2014). [CrossRef]  

23. M. Saruwatari and S. Kawanishi, “Ultrawide frequency response measurement system using optical heterodyne detection method,” in 1988 Conference on Precision Electromagnetic Measurements. (IEEE, 1998), pp. 403–405.

24. Z. Zhenghua, Y. Chun, C. Zhewei, C. Yuhua, and L. Xianghua, “An ultra-low phase noise and highly stable optoelectronic oscillator utilizing IL-PLL,” IEEE Photonics Technol. Lett. 28(4), 516–519 (2016). [CrossRef]  

25. J. Tang, T. Hao, W. Li, D. Domenech, R. Baños, P. Muñoz, N. Zhu, J. Capmany, and M. Li, “Integrated optoelectronic oscillator,” Opt. Express 26(9), 12257–12265 (2018). [CrossRef]  

26. K.-H. Lee, J.-Y. Kim, and W.-Y. Choi, “Injection-locked hybrid optoelectronic oscillators for single-mode oscillation,” IEEE Photonics Technol. Lett. 20(19), 1645–1647 (2008). [CrossRef]  

27. R. Fu, X. Jin, Y. Zhu, X. Jin, X. Yu, S. Zheng, H. Chi, and X. Zhang, “Frequency stability optimization of an OEO using phase-locked-loop and self-injection-locking,” Opt. Commun. 386, 27–30 (2017). [CrossRef]  

28. X. Zhang, X. Zhou, and A. S. Daryoush, “A theoretical and experimental study of the noise behavior of subharmonically injection locked local oscillators,” IEEE Trans. Microwave Theory Tech. 40(5), 895–902 (1992). [CrossRef]  

29. A. Neyer and E. Voges, “Dynamics of electrooptic bistable devices with delayed feedback,” IEEE J. Quantum Electron. 18(12), 2009–2015 (1982). [CrossRef]  

30. M. J. Feigenbaum, “Quantitative universality for a class of nonlinear Transformations,” J. Stat. Phys. 19(1), 25–52 (1978). [CrossRef]  

31. M. J. Feigenbaum, “Universal behavior in nonlinear systems,” Phys. D 7(1-3), 16–39 (1983). [CrossRef]  

32. P. Collet and J.-P. Eckmann, Iterated maps on the interval as dynamical systems (Springer Science & Business Media, 2009).

33. L. Larger, P.-A. Lacourt, S. Poinsot, and M. Hanna, “From flow to map in an experimental high-dimensional electro-optic nonlinear delay oscillator,” Phys. Rev. Lett. 95(4), 043903 (2005). [CrossRef]  

34. V. J. Urick, K. J. Williams, and J. D. McKinney, Fundamentals of microwave photonics (John Wiley & Sons, 2015).