1. Introduction

Optoelectronic oscillators (OEOs) are microwave photonic systems that have found a plethora of applications in various disciplines related to microwave photonics (see review article [1]). The most important and most investigated application so far is the generation of ultra-pure radiofrequency (RF) signals [2–4]. In their most basic configuration, they feature a closed loop where an amplitude-modulated laser signal is launched into a few km-long optical fiber delay line before being photodetected, and the output microwave signal is frequency-filtered and amplified before being fed back to the light modulator [5]. This original architecture had the great advantage of simplicity and offers a competitive phase noise performance. However, since the filtering occurs in the RF domain, the resulting oscillator is generally not tunable in frequency. Another drawback is narrowband RF filtering in the mm-wave range (frequency beyond $30$ GHz) becomes progressively less effective as the frequency increases.

Both these problems are addressed by OEOs based on stimulated Brillouin scattering (SBS). Using Brillouin scattering to improve the functionalities of OEOs is a widespread strategy that has been implemented in various platforms [6–9]. In the configuration of interest in this article, a pump laser is launched in an highly nonlinear fiber spool in order to generate a Brillouin gain profile at an adjacent redshifted frequency. This process permits the selective amplification of a phase-modulated and contra-propagating laser signal. Most importantly, the RF oscillation frequency solely depends on the spectral gap between the pump and signal lasers, and is thereby easily tunable. From that regard, a main characteristic of this approach is that frequency selection is achieved by the means of a microwave photonic filter instead of an RF filter [10–12]. Once the RF beatnote is generated after photodetection, it is filtered, amplified and fed back to the phase modulator to close the optoelectronic loop. It has been demonstrated by several research groups that these SBS-based OEOs feature frequency tunability from DC to more than $60$ GHz and an ultra-low phase noise figure [13–15]. In fact, the km-long fiber spool can be replaced by a sub-cm-long highly nonlinear waveguide, and in that case, the system has the potential to be reduced to a chipscale size [16].

 

Fig. 1. Schematic representation of the SBS-based OEO, for the generic case where $T \neq 0$. Circ: Circulator; Filter: RF filter intended to select the beatnote frequency $\Omega _{\mathrm {rf}}$ (wideband); ISO: Isolator; HNLF: Highly nonlinear fiber; PC: Polarization-controller; PD: Photodetector; PM: Phase modulator; PS: Phase shifter; RFA: Radiofrequency amplifier.

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Despite the potential and many distinctive advantages of SBS-based OEOs, no dynamical model has ever been proposed to describe their nonlinear behavior. Indeed, previous research has shown that it is in general possible to obtain an equation for the envelope of the microwave generated by these OEOs [1,17–19]. In this article, we undertake a theoretical analysis showing that an delay-differential envelope equation for SBS-OEOs can be derived, analogously to what had been done earlier for the conventional architectures of OEOs. In agreement is shown that as the gain is increased, the system undergoes a primary Hopf bifurcation and then a secondary Neimark-Sacker bifurcation before being driven to fully developed chaos.

The outline of the article paper is following. In Sec. 2, the SBS-based OEO is presented and its principle of operation is described. Section 3 is devoted to the analysis of the stimulated Brillouin scattering dynamics, and an input-output relationship is derived for the selectively amplified signal wave. The dynamical characterization of the SBS-based microwave photonic filter enables us to obtain a time-domain differential equation for the OEO in Sec. 4. Sections 5 and 6 are focused on evaluating the stability of the stationary dynamical states predicted by the model in the cases where the frequency-selective amplification is mediated by a km-long fiber delay line ($T\neq 0$) or by a sub-cm-long waveguide ($T \simeq 0$), respectively. The last section concludes the article.

2. System and operation principle

The schematic representation of the SBS-based OEO is presented in Fig. 1. The dynamics of this system can be described by two key variables. The first one relates to electric field $E(t)$ at the input of the phase modulator, while the second relates to the voltage $V(t)$ feeding the modulator. Instead of working directly with the real-valued variables $E(t)$ and $V(t)$, we use instead their complex slowly-varying amplitudes ${\cal E}(t)=E(t) e^{i\varphi (t)}$ and ${\cal V}(t)=V(t) e^{i\psi (t)}$ defined through

 

Fig. 2. Schematic Representation of a chipscale SBS-based OEO. This system corresponds to the case $T=0$ in our analysis (see Sec. 4.4). Only the essential elements have been included in the figure. Circ: Circulator; HNLW: Highly nonlinear waveguide; PD: Photodetector. PM: Phase modulator; A proof-of-concept experiment for chispscale SBS-based OEOs has been presented in Ref. [16], using a chalcogenide waveguide.

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The pump optical path is also called high-order frequency selection branch, and it is where the RF frequency $\Omega _0$ is determined. The pump field is $E_p(t)$ while $E_{\mathrm {s}}(t)$ is the phase-modulated signal field that is launched at the input of the highly nonlinear fiber. The two waves counter-propagate in the nonlinear fiber spool, and the pump wave stimulates Brillouin backscattering. The frequencies of the fields involved in this process are related as $\omega _{\mathrm {p}} = \omega _{\mathrm {s}} +\Omega _{\mathrm {rf}} + \Omega _{\mathrm {B}}$, so that the RF output frequency can be obtained as

It is important to note that one of the most outstanding challenge in OEO technology is the generation of mm-waves [20–23]. In general, achieving such high frequencies ($>30$ GHz) is achieved via frequency multiplication [24–30]. Additionally, The other outstanding challenge for OEOs is tunability, which has been addressed using several different techniques [31–40]. From that perspective, a key advantage of SBS-based OEOs is that they addresses both challenges simultaneously and the only limitation for the maximal frequency of the RF signal is the bandwidth of the photodiode and phase modulator.

3. Frequency-selective amplification based on stimulated Brillouin scattering

3.1 Brillouin amplification

The phenomenology of frequency-selective Brillouin amplification is the following. We consider that the highly nonlinear fiber of length $L$ and aligned along an axis $z$ such that the fiber is located at $0 \leq z \leq L$. The pump laser wave

Consider now a contra-propagating signal laser beam $E_{\mathrm {s}}(t)$ defined as

3.2 Equations of the traveling waves

The time-domain slowly-varying envelope of the input and output signals are written as ${\mathcal {E}}_{\mathrm {s}}^{\mathrm {in}}(t)$ and ${\mathcal {E}}_{\mathrm {s}}^{\mathrm {out}}(t)$, respectively (in units of $\sqrt {\mathrm {W}}$). We can introduce their corresponding intensities as ${I}_{\mathrm {s}}^{\mathrm {in,out}}(t) = |{\mathcal {E}}_{\mathrm {s}}^{\mathrm {in,out}}(t)|^{2}/A_{\mathrm {eff}}$ (in units of W/m$^{2}$), with $A_{\mathrm {eff}}$ being the effective area of the fiber.

If we neglect dispersion and cross/self-phase modulation in the fiber, the pump and signal intensities ${{I}}_{\mathrm {p}}(z,t)$ and ${{I}}_{\mathrm {s}}(z,t)$ at any point of the Brillouin amplifier obey the well-known equations [41,42]

We furthermore make the following two assumptions: (i) The pump laser power is assumed undepleted (i. e. ${{I}}_{\mathrm {p}}$ is assumed approximately constant, and such that ${{I}}_{\mathrm {p}} \gg {{I}}_{\mathrm {s}}$); (ii) and the losses in the fiber are negligible in comparison to the Brillouin amplification (i. e., $\alpha \simeq 0$). As a consequence, the above equations are reduced to the single equation

3.3 Accounting for frequency selection

Equation (13) does not account for frequency selection in the amplification process; more precisely, it assumes that the pump and the input/output signals are perfectly monochromatic signals with center frequencies $\omega _{\mathrm {p}}$ and $\omega _{\mathrm {s}}=\omega _{\mathrm {B}}$, respectively.

Accounting for the effect of the Brillouin gain bandwidth on the amplification process requires rewrite Eq. (13) in the Fourier domain and to replace the gain parameter $g_{_{\mathrm {B0}}}$ by the Brillouin gain profile $g_{_{\mathrm {B}}}(\Omega )$, following

We need now to rewrite Eq. (14) in terms of fields in the Fourier domain, so that the phase information of all the signals involved can be retrieved. We first express the Brillouin gain as

Because of this useful mathematical property (the possibility to perform an inverse Fourier transform), we propose to generalize the small-gain expansion of Eq. (19) to the case of large gain. We therefore consider that gain is the sum of a flat background ($=1$) and a Lorentzian, even in the case of large gain, that is

For the approximation of Eq. (20) to be physically valid, we have to make sure that the peak value and half-linewidth values of the equivalent Lorentzian gain $1+{G}_{\mathrm {o}} \mathcal {L}_{\mathrm {eq}}(\Omega )$ match those of the “exact” profile $e^{\frac {1}{2}g_{_{\mathrm {B0}}}{I}_{\mathrm {p}} L \, \mathcal {L}_{\mathrm {B}}(\Omega )}$. We therefore have the equalities

3.4 Time-domain equations for Brillouin amplification

We now have all the elements needed to perform the inverse Fourier transform that will give the us the equations ruling the field dynamics in the time domain. Let us first rewrite the output signal as

It is interesting to note that: (i) We do not need to add a detuning $\sigma$; (ii) We adopt an input-output convention such that the excitation term is not multiplied by $i$; (iii) By definition, the initial condition for ${\mathcal {E}}_{\mathrm {s}}^{\mathrm {amp}}$ is null, i.e. ${\mathcal {E}}_{\mathrm {s}}^{\mathrm {amp}}(0)=0$.

4. OEO model

4.1 Phase modulation

The signal laser emits a monochromatic wave, that can be written as

The first sidemode ${\cal E}_1(t)$ is now injected into the Brillouin amplifier. Following the analysis led in the preceding Section, the amplified portion of the signal in the fiber obeys the equation

4.2 Accounting for the Brillouin amplification

Now that we have the equation ruling the Brillouin amplification, we can proceed with calculating the voltage generated by the photodetection of this optical signal, which can be expressed as (in units of V):

We now need to put all these elements together in order to obtain our final model. Let us introduce the dimensionless microwave envelope

We therefore have the following four-step model for our tunable OEO based on Brillouin amplification:

Note that the gain of this OEO can be tuned in this system following two different ways: either via the optical gain $G_\textrm {o}$ (controlled by the power of the pump laser), or via the optoelectronic gain $\beta$ (controlled via the gain $G_\textrm {e}$ of the RF amplifier of the electronic branch). In this study, we will consider without loss of generality that the optical gain $G_\textrm {o}$ is fixed, while the optoelectronic gain $\beta$ is tunable via $G_\textrm {e}$.

4.3 Envelope equation for the generic case $T \neq 0$

The four-step model presented in the preceding subsection can be simplified. We have indeed

Let us now rewrite Eqs. (54) and (50) as

We can now determine $\dot {{\cal A}}$ as

Equation (64) is well-defined, and in particular, the time-delayed terms in the brackets in the right-hand side are well-defined at any time. The term $\partial _t {|{\cal A}_T|}$ is indeed unusual, but is not problematic from the mathematical viewpoint because at any time $t$, the variable ${\cal A}_T$ and any variable that univocally depends on it can be determined unambiguously, even at the numerical level. One can also note that in the steady state, we have $\dot {{\cal A}}=0$, $\partial _t {|{\cal A}_T|}=0$. Note that unlike the conventional envelope equations for OEOs, Eq. (64) involves coupling between the delayed and non-delayed variables. Also note that the phase dynamics is irrelevant in the deterministic case, and can be disregarded but will become key at the time to investigate the noise performance of the system [45–49].

4.4 Envelope equation for the particular case $T = 0$

Using Eq. (64), we can obtain an envelope equation for the case $T=0$, which corresponds to the physical configuration where the system is chipscale (see Fig. 2). We first note that the phase $\psi$ of the microwave envelope ${\cal A}$ is a constant of motion [see Eq. (53)]. It results that $[\partial _t {|{\cal A}|}] \, {\cal A} = \dot {\cal A} |{\cal A}|$ and consequently, Eq. (64) can be written as

5. Stationary states and their stability for $T = 0$

We already know that the phase of ${\cal A}$ is an arbitrary constant, which can be set without loss of generality to zero. In that case, ${\cal A}$ becomes real-valued and we can remove the calligraphic fonts following ${\cal A} \equiv {A} \geq 0$, so that the amplitude equation reads

 

Fig. 3. The graphical representation of Eq. (68) where $\mathrm {L}({{A}_\textrm {st}})={{A}_\textrm {st}}/{4\Gamma }$ (linear in ${A_\textrm {st}}$) and $\mathrm {NL}({{A}_\textrm {st}})={\textrm {J}_1^{2} [ {A}_\textrm {st} ] }/{A_\textrm {st}}$ (nonlinear in ${A_\textrm {st}}$) are expressed as discontinuous colored and continuous black lines respectively.

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Equation (69) shows two possible oscillatory solutions, but we can reduce to one by confirming the limit of the Bessel-Cardinal function. The function oscillates between $-0.066<\textrm {Jc}_{1}(x)<0.5$. If we consider $\Gamma$ is a positive value, the two solutions should cover the positive and negative range separately, and thus

5.1 Stability of the trivial solution

To evaluate the stability of the trivial solution $A_\textrm {tr}=0$, we can perturb the fixed point and check if it decays or grows by depending on the loop gain $\Gamma$. Considering a very small perturbation $\delta A$ around the fixed point leads Eq. (67) to be written as

5.2 Stability of the oscillating solution

We now perturb the oscillating solution $A_\textrm {osc}$ (which only exists for $\Gamma >1$), so that the linearized counterpart of Eq. (67) can be written as

5.3 Numerical simulation for the case $T=0$

Figures 4(a) and (b) display the transient dynamics of the SBS-based OEO with null delay for $\Gamma =0.9$ and $\Gamma =1.1$, that is, just below and above the threshold value $\Gamma _\textrm {th}=1$. On the other hand, Figs. 4(c) and (d) show the dynamics of the oscillator at $\Gamma =2.49$ and $\Gamma =56$, thereby showing that oscillating solution is indeed stable in the gain range of interest in for that solution ($1<\Gamma <56$). The amplitudes of the oscillating solutions also correspond to the intersections points in Fig. 3.

 

Fig. 4. Numerical simulation of the SBS-based OEO with null delay [Eq. (66)] for various values of the gain $\Gamma$. The value of the Brillouin gain bandwidth is set to $\mu = 2\pi \times 25$ MHz.

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6. Stationary states and their stability for $T \neq 0$

We now consider the generic case where the delay is not null. Here also we can take advantage of the fact that that the phase $\psi$ of the microwave is a constant. Once again it is convenient to arbitrarily set $\psi =0$, so that the envelope ${\cal A}$ becomes real-valued, with ${\cal A} \equiv A >0$ and ${\cal A}_T \equiv A_T >0$. The microwave dynamics is now ruled by

We can determine the stability of the two solutions in the same way as $T=0$ case.

6.1 Stability of the trivial solution

When we perturb the trivial solution with $\delta A=\delta A_0 \exp [{(\lambda _\textrm {tr}+i\xi _\textrm {tr})t}]$, the coefficients $\lambda _\textrm {tr}$ and the $\xi _\textrm {tr}$ will obey

6.2 Stability of the oscillating solution

When we perturb the oscillatory solution $A_{\mathrm {osc}}$, the linearization of Eq. (74) yields

The secondary bifurcation occurring at $\Gamma _\textrm {cr} = 2.49$ is a Neimark-Sacker or torus bifurcation [1,17,19,50,51]. It should be noted that in the case of a short (but non-null) delay with $\mu T \sim 1$, the formalism developed above is not valid anymore and a more involved calculation would be needed to determine $\Gamma _\textrm {cr}$ – see Ref. [52].

6.3 Numerical simulation for the case $T\neq 0$

Figures 5(a) and (b) display the transient dynamics of $\Gamma =0.9$ and $\Gamma =1.1$ the just below and above the threshold value $\Gamma _\textrm {th}$. On the other hand, Figs. 5(c) and (d) show the dynamics of the oscillator at $\Gamma =2.48$ and $\Gamma =2.5$, that is, just before and just after the torus bifurcation. The amplitude of the output microwave is stable below the critical value ($\Gamma _\textrm {cr}$) and unstable above, as predicted by the bifurcation analysis. When the gain is further increased, the oscillator displays $4T$-periodic behavior [Fig. 5(e)] while further increase of $\Gamma$ leads to a chaotic behavior. [Fig. 5(f)]. The overall dynamics of the oscillator can be described via a bifurcation diagram as displayed in Fig. 6, where the Hopf and Neimark-Sacker bifurcations are shown to occur exactly where they where analytically predicted.

 

Fig. 5. Numerical simulation of the SBS-based OEO when $T \neq 0$ [Eq. (74)] for various values of the gain $\Gamma$. The insets of (c) and (d) show the asymptotic dynamical behavior of the system, evidencing the qualitative difference induced by the Neimark-Sacker bifurcation at $\Gamma _{\mathrm {cr}}=2.49$. The parameters of the system are $\mu = 2\pi \times 25$ MHz and $T=5$ $\mu$s.

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Fig. 6. Bifurcation diagram for the SBS-based OEO. The labels (a)–(f) indicate the values of the gain $\Gamma$ used for the time-domain simulations in Fig. 5. The Hopf and Neimark-Sacker bifurcations numerically emerge at the gain values $\Gamma _{\mathrm {th}}=1$ and $\Gamma _{\mathrm {cr}}=2.49$ (resp.), as predicted by the theory. The parameters of the system are $\mu = 2\pi \times 25$ MHz and $T=5$ $\mu$s.

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7. Conclusion

In this article, we have studied the nonlinear dynamics of SBS-based OEOs. We have first analyzed the SBS effect in the nonlinear fiber acting as a microwave photonic filter, and we have derived as well the differential equation ruling the input-output relationship of frequency-selective amplification. This modelling of the SBS process has enabled us to obtain a delay-differential equation governing the microwave envelope dynamics of the OEO output signal. We have then performed a stationary state analysis in order to find the various solutions (trivial and oscillatory), and to evaluate their stability. This procedure has led us to the analytical determination of the threshold gain value triggering the onset of microwave oscillations, as well as the critical value leading to their destabilization via amplitude modulation. Our numerical simulations have confirmed the stability analysis.

Future research will specifically focus on the phase noise optimization of these oscillations. This endeavor will include a detailed analysis of the beneficial effects of Brillouin scattering with regard to the close-in phase noise performance of the output microwave.