1. Introduction

Time delays appear naturally in a variety of physical, biological, chemical and technological systems, and are a source of oscillatory instabilities [1]. When the delay is large compared to the other time scales of the system, square-waves are the dominant solutions for specific values of the parameters. Motivated by applications such as optical clocks [2] or optical sensing [3], the generation of stable pulses with square-wave form has been investigated in optical and optoelectronic systems during the past few years [2–10].

The possibility to generate square-wave pulses of controllable period has been discussed in the context of the Ikeda differential equation [11–13], which presents a nonlinear feedback term proportional to cos[x(t − τ) + Φ], where τ is the delay time and Φ a constant offset phase which fixes the operating point, namely, depending on the value of Φ the effect of the feedback term is to damp small perturbations (negative feedback) or to amplify them (positive feedback). The Ikeda equation was originally introduced to describe the dynamics of the transmitted light from a nonlinear absorbing medium subject to delayed optical feedback [14] and has been later generalized to describe other systems with delayed feedback [15, 16], among them, optoelectronic oscillators (OEOs). While several OEO configurations have been considered [17, 19], including those based on whispering gallery modes [18] and multicore fibers [20], here we focus on delay-line OEOs [21] which can be modeled by an Ikeda-like equation with feedback proportional to cos²[x(t − τ) + Φ] [22]. Feedback is negative for offset phases Φ [0, ∈ π/2] and positive for Φ ∈ [−π/2, 0]. Square waves originated with negative feedback have a symmetric duty cycle, namely the duty cycle is half the period. In contrast, for positive feedback square waves have an asymmetric duty cycle. Generation of stable square waves with symmetric and asymmetric duty cycles in OEOs has been reported in Refs. [23–25]. Stable square waves have also been found in other optical systems such as edge-emitting diode lasers [4–6], vertical cavity surface-emitting lasers [7,8,26], ring lasers [9] or mode-locked fiber lasers [10,27].

Coupling with delay two oscillators, each with an intrinsic long delay so that it generates square-waves, leads to interesting dynamics arising from the interplay between the intrinsic and the coupling delay times. In particular, two mutually coupled identical OEOs with negative feedback can generate stable in-phase and out-of-phase synchronized square waves with symmetric duty cycle when the ratio between the delay times fulfills suitable conditions [28]. For positive feedback in- and out-of-phase synchronization is also obtained but in this case the duty cycle is not symmetric [29]. In general, the square-waves generated so far with optical and optoelectronic systems have been obtained synchronized in phase or out of phase. In this work we show that two mutually coupled non-identical OEOs, besides in-phase and out-of-phase, can generate square-waves synchronized at a quarter of the period (T/4).

Synchronization at a quarter of the period has been observed in some animal gaits of quadrupeds as walk and jump [30], in the limb coordination in crustacean swimming [31], between the oscillations of pedestrian walking on a bridge and the structural oscillation [32], in predator-prey cycles [33–35], in human cortical sources [36], neural networks [37] and in solar transverse oscillations [38]. Nevertheless, in none of these studies the oscillations have a square-wave shape. Here by exploring the dynamical regimes that can arise in a prototypical model for mutually delay-coupled OEOs we show that stable square-wave pulses synchronized at a quarter of the period do exist in a broad parameter region. The key point to obtain such T/4 solutions is that the two OEO operate with different offset phases, in such a way that the feedback is negative in one and positive in the other, namely the feedback is mixed. It should also be emphasized that these T/4 solutions are in the ns time scale, orders of magnitude faster than the above systems.

The outline of the article is as follows. We describe the model in Sec. 2. In Sec. 3 we study the Hopf instabilities of the steady state, in Sec. 4 the in- and out-of-phase square waves with symmetric duty cycle occurring when both OEOs have negative feedback, in Sec. 5 the periodic solutions with asymmetric duty cycle arising when both OEOs have positive feedback, in Sec. 6 the square waves synchronized at a quarter of the period, and in Sec. 7 the effect of a small mismatch in the delay times on the degradation of the T/4 solutions. Sec. 8 concludes the paper.

2. System description

We consider two OEOs, similar to those of [39], mutually coupled with delay, both fed by half of the light from a cw semiconductor laser (LD) emitting with intensity P as illustrated in Fig. 1. Each OEO consists of a Mach-Zehnder interferometer (MZI), an optical delay line, a photodiode (PD) with sensitivity S and an amplifier. The variables related to loop i are identified with subindex i, i = 1, 2. The optical power after MZI_i is split into two parts. A fraction α_ii is delayed T_ii within OEO_i (self-feedback) and a fraction α_ij with j ≠ i couples OEO_i with OEOj with a delay time T_ij (cross-feedback). A photodiode (PD) detects the intensity resulting from the combination of the self- and cross-feedback optical beams and produces an electrical signal, which is amplified and band-pass filtered, and finally used to drive the Mach-Zehnder AC electrode. Assuming no reflections in the optical paths, we can take the optical electric field as a scalar. We also consider that the voltage V_i applied to MZI_i has a DC and a RF component, $V_{B_i}$ and $V_{R{F}_i}(t)$ respectively, and introduce the dimensionless variables $x_i(t)=\pi V_{R{F}_i}(t)/2V_{{\pi }_i}$ and ${\mathrm{\Phi }}_i=\pi V_{B_i}/2V_{{\pi }_i}$, being V_π the voltage required for a change of π in the phase. Introducing two additional variables, $y_i(t)={\displaystyle {\int }_{t_0}^t{x}_i(t^{\prime })d{t}^{\prime }}$, the dynamics of the system is ruled by [28],

 

Fig. 1 Diagram of the system modeled integrated by two mutually delay-coupled OEOs. Each OEO consists of a Mach-Zehnder interferometer (MZI), a fiber loop with delay time T_f, a photodiode (PD) and a RF amplifier (G) whose output modulates an arm of the MZI. OEOs are fed by a laser diode (LD) whose output is split in two parts by a 50/50 fiber splitter. The OEOs are mutually coupled with cross-feedback delay time T_c.

Download Full Size | PDF

Introducing $Y_i(t)=[y_i(t)-y_i^{\text{st}}]/T_c$ and scaling the time with T_c, s = t/T_c, we get

3. Oscillatory instabilities of the steady state

Equations (3) admit periodic square-wave solutions in the limit of large delays which are born as sinusoidal solutions from a Hopf bifurcation of the steady state $x_i^{\text{st}}=Y_i^{\text{st}}=0$ which takes place increasing P. Thus we first analyze the steady state stability by considering the linearized equations for the small perturbations, $U_i(s)=x_i(s)-x_i^{\text{st}}$ and $V_i(s)=Y_i(s)-Y_i^{\text{st}}$,

Equations (5) admit solutions of exponential form, U_i(s) = u_i exp[(λ + iω)s] and, since V_i is the integral of U_i, V_i(s) = u_i exp[(λ + iω)s]/(λ + iω). Replacing U_i and V_i in Eqs. (5) leads to

The condition for non-trivial solutions is that the determinant vanishes, namely

The steady state becomes unstable at the bifurcation point P = P_c where λ = 0. Splitting Eq. (8) into real and imaginary parts we obtain for λ = 0:

The solutions of Eq. (9) can be seen as the zeros of a second order polynomial for P_c whose coefficients depend on the system parameters and the perturbation frequency ω. We first consider the case ε = δ = 0. The value of the polynomial is 1 for P_c = 0, thus for P_c small Eq. (9) has no solutions meaning that the steady state is stable. For given F_i, K_i and s₀, the first instabilities are those such that the linear coefficient, (F₁ +F₂)cos(ωs₀), takes the most negative value, namely

For Eqs. (12) and (11) to be mutually compatible it is required that,

Thus the instabilities with lower threshold take place for s₀ rational. The period of the emerging oscillatory solutions is given by

The Hopf instability threshold, P_c, can be obtained substituting Eqs. (12) and (11) in Eq. (9): else the smaller positive solution for P_c is

If (F₁ F₂) = K₁K₂(−1)^k than

In order Eq. (17) to be real it is required that (F₁ − F₂)² + 4K₁K₂(−1)^k > 0. Since P_c depends only on the parity of k′ and k many bifurcations have the same threshold.

Introducing Eqs. (12) and (11) in Eq. (7) leads to a ratio for the perturbation amplitudes:

For k even, Q is real and there are two kinds of solutions depending on the sign of Q. For Q > 0 u₁ and u₂ have the same sign and the instability leads to oscillatory solutions in which x₁ and x₂ are in phase (although the amplitude may not be the same) while for Q < 0 the instability leads to out-of-phase oscillations. For k odd, Q is a purely imaginary number. Considering x_i = U_i + c.c. this leads to x₁ and x₂ being dephased by T/4.

In Fig. 2 we plot P_c obtained from Eqs. (16) and (17) for γ_ij = γ = 0.5 and different parities of k and k′. Provided γ_ij are identical, the same results are obtained for other values of γ rescaling P_c with γ². Pink regions correspond to offset phases for which P_c > 5, while white and grey regions correspond to unphysical solutions for P_c (negative or complex values respectively). For Φ₁ ≃ Φ₂, along the diagonal, only instabilities associated to k even can take place [panels (a) and (b)]. Thus, as pump is increased, the steady state becomes unstable to either in-phase or out-of-phase oscillations. On the contrary, for Φ₁ = −Φ₂, along the anti-diagonal, the only possible instabilities are those with k odd [panels (c) and (d)] leading to periodic oscillations dephased by T/4. In fact, there is a broad parameter region where the lower threshold corresponds to k-odd instabilities leading to T/4 dephased oscillations.

 

Fig. 2 P_c as given by Eqs. (16)–(17) for γ_ij = 0.5 and Φ₀ = 0. (a) k′ odd and k even, (b) k′ even and k even, (c) k′ odd and k odd, (d) k′ even and k odd. Parameter regions in which P_c is negative or imaginary are plotted in white and grey, respectively.

Download Full Size | PDF

For non-identical γ_ij regions in (Φ₁,Φ₂) parameter space get distorted as illustrated in Fig. 3. Nevertheless there are regions where k-even instabilities dominate, such as Φ₁ ≈ Φ₂, leading to in-phase and out-of-phase solutions, and regions where k-odd instabilities leading to T/4 dephased solutions are the first to take place increasing the pump. In general, physical solutions for P_c with k′ odd and k even [panel (a) in Figs. 2 and 3] exist for offset phases such that F₁ + F₂ > 0 (negative feedback), solutions with k′ even and k [panel (b)] odd exist for offset phases such that F₁ + F₂ < 0 (positive feedback), and solutions with k odd [panels (c) and (d)] exist for offset phases such that K₁K₂ < 0 (mixed feedback).

 

Fig. 3 Value of P_c as in Fig. 2 for γ₁₁ = 0.5, γ₂₂ = 0.3, γ₁₂ = 0.2 and γ₂₁ = 0.4.

Download Full Size | PDF

We note that for Φ₀ = 0, changing the sign of Φ₁ and Φ₂ changes the sign of F_i and K_i so that Eqs. (16) and (17) are the same replacing k′ by k′ + 1. As a consequence, the instabilities associated to k′ odd occur for offset phases which are the specular image of those for k′ even [compare panel (a) with (b) and (c) with (d) in Figs. 2 and 3]. This symmetry is broken for Φ₀ ≠ 0.

In the general case of ε ≠ 0, δ ≠ 0 and arbitrary s₀ we can determine P_c and ω solving numerically Eqs. (9) and (10). From Eq. (7) we can also obtain a theoretical prediction for Q = u₂/u₁, which is a complex number whose modulus,

In the next sections we analyze the in- and out-of-phase solutions arising for negative and positive feedback, and the T/4 dephased solutions for mixed feedback.

4. Onset of periodic square waves for negative feedback

As discussed in Ref. [28] for identical γ_ij, Φ₀ = 0 and Φ₁ = Φ₂ = Φ ∈ [0,π/2], increasing the pump the steady state becomes unstable to in-phase or out-of-phase periodic oscillations. The instability is supercritical and, as the pump is further increased, the oscillations soon become square-shape with a duty cycle half of the period. Owing to parameter symmetry the oscillations in x₁ and x₂ have the same amplitude. Here we show that square waves with symmetric duty cycle can also be found for non-identical γ_ij or offset phases (although x₁ and x₂ have different amplitudes) in a broad parameter region associated to negative feedback in both OEOs. More precisely they arise for parameter values such that the steady state instabilities associated to k′ odd and k even dominate [compare panel (a) in Figs. 2 or 3 with the other panels].

To illustrate this we consider Φ₁ = 0.2π, Φ₂ = 0.3π and γ_ij = 0.5. Fig. 4 shows all the Hopf bifurcations in the range ω < 10π and 0 < P_c < 2 obtained solving numerically Eqs. (9) and (10). For these parameters, Eq. (17) has physical solutions only for k′ odd and k even (See Fig. 2). Thus, the bifurcations with lower threshold occur at s₀ = 2k′/k = k′/n (n = k/2 integer), namely at rational fractions s₀ = q′/q (irreducible) with odd numerator. At these values of s₀ a set of bifurcation lines with k′ = jq′ and k = 2jq (j odd) are born simultaneously [Fig. 4(a)]. j = 1 corresponds to the fundamental solution with ω = qπ while j = 3,5,… correspond to harmonics with ω = jqπ. Moving away from s₀ = q′/q the degeneracy is broken and P_c has a parabolic dependence with s₀ − q′/q, the parabola being narrower for higher harmonics. For q odd, k = 2jq is even but not multiple of 4 (singly even) and Q > 0, so that all the harmonics are in phase (plotted in red). Conversely, for q even, k = 2jq is multiple of 4 (doubly even) and Q < 0, which leads to out-of-phase oscillations (black lines).

 

Fig. 4 Hopf bifurcations with ω < 10π for Φ₁ = 0.2π, Φ₂ = 0.3π, Φ₀ = 0, γ_ij = 0.5, ε = 4.17 × 10⁻⁴ and δ = 1.2 × 10⁻². In (b) lines are plotted only in the range where P_c < 2. Red and black lines correspond to in- and out-of-phase oscillations, respectively.

Download Full Size | PDF

The previous results have been obtained from the linear stability analysis of the steady state. To analyze to which extend this results are valid far away from the bifurcation point we resource to numerical integration of the full nonlinear dynamical equations (3). Figure 5 shows the fundamental out-of-phase solution for Φ₁ = 0.2π and Φ₂ = 0.25π and s₀ = 3/4, so that k′ = 3 and k = 8. For the parameters of the figure the steady state becomes unstable at P_c = 2.1167, giving rise to out-of-phase quasi-sinusoidal oscillations [Fig. 5(a)] with a period T = 1/2 (20ns) as predicted by Eq. (14). While the two OEOs are synchronized in frequency, amplitudes differ because each oscillator receives a different feedback strength. The ratio Q between the amplitudes of the oscillations in x₁ and x₂ obtained from the numerical integration of (3), Q = −0.2529, is in good agreement with the prediction of the linear stability analysis (19), Q = 0−.2521, and also with the approximation (18), Q = −0.2519. Increasing the pump the shape of the solution becomes less and less sinusoidal approaching a square-wave [Figs. 5(b) and 5(c)]. Nevertheless the solutions remain synchronized out of phase and the ratio between the amplitudes Q remains practically constant. Similar good agreements are found for in-phase solutions arising at delay times fulfilling the ratio s₀ = q′/q with q′ and q odd.

 

Fig. 5 Out-of-phase oscillations with symmetric duty cycle for Φ₁ = 0.2π, Φ₂ = 0.25π, Φ₀ = 0, γ₁₁ = 0.5, γ₂₂ = 0.3, γ₁₂ = 0.2, γ₂₁ = 0.4, T_f = 30ns, T_c = 40ns, ε = 6.25 × 10⁻⁴, δ = 8 × 10⁻³ and P = 2.117 (a), P = 2.12 (b), and P = 2.3 (c). Black and red lines correspond to x₁ and x₂ respectively.

Download Full Size | PDF

5. Onset of periodic oscillations for positive feedback

For Φ₀ = 0 and Φ₁ = Φ₂ = Φ ∈ [−π/2,0] square waves typically have an asymmetric duty cycle and are born subcritically [29]. The degree of asymmetry depends on the offset phase, so that the duty cycle is symmetric (and square waves are supercritical) for Φ = −π/4 and becomes progressively more asymmetric as the offset phase is changed away form −π/4. For delay times whose ratio is rational, s₀ = q′/q irreducible, for q′ odd the system displays in- phase square waves while for q′ even both in-phase and out-of-phase asymmetric square waves coexist. The system can also display periodic oscillations with a period much longer than the delay times, typically in the microsecond regime. These oscillations are always in phase, born supercritically and do not have a square-wave shape. We show here that a similar scenario occurs for non-identical offset phases and parameter values for which instabilities associated to k′ and k both even dominate [compare (b) in Figs. 2 or 3 with the other panels].

To illustrate this we consider Φ₁ = 0.2π, Φ₂ = 0.3π and γ_ij = 0.5. Figure 6 shows all the Hopf bifurcations in the range ω < 10π and 0 < P_c < 1.15 from numerical solution of Eqs. (9) and (10). For these parameters, Eq. (17) has physical solutions only for k′ odd and k even (See Fig. 2). A set of Hopf bifurcation lines are born simultaneously at rational values for the delay ratio s₀ = q′/q, irreducible. Different Hopf lines correspond to different values of k′ = jq′ and k = 2jq. For q′ odd, j must even, thus k is doubly even and Q > 0, which leads to in-phase oscillations. For q′ even any j is allowed (q′/q irreducibility requires q odd which is not restrictive since k = 2jq is always even). The fundamental solution j = 1 and the odd harmonics have a singly even k and Q < 0 and thus are out of phase, while the even harmonics are in phase. As before degeneracy is broken for s₀ different from q′/q and Hopf lines have a parabolic shape which is narrower for higher harmonics.

 

Fig. 6 Hopf bifurcations for Φ₁ = −0.2π and Φ₂ = −0.3π. Other parameters as in Fig. 4. The green line corresponds to microsecond oscillations.

Download Full Size | PDF

For identical γ_ij and Φ₀ = 0 square waves have a symmetric duty cycle and arise supercritically if Φ₁ +Φ₂ = −π/2, as the fundamental out-of-phase solution for T_f = 40ns and T_c = 60ns (s₀ = 2/3) shown in Fig. 7. Here k′ = 2, k = 6 and the solution has a period T = 2/3 (40ns). For other offset phases oscillations are born subcritically and have an asymmetric duty cycle, as shown in Fig. 8 for the same delay times but with non-identical γ_ij, Φ₁ = −0.25π, and Φ₂ = −0.15π. For the parameters of Fig. 8, P_c = 1.6334. As pump is increased above threshold the periodic solution becomes more square-wave shaped while the duty cycle asymmetry decreases [Figs. 8(c) and 8(d)]. Below threshold, decreasing pump the duty cycle asymmetry increases until the length of the shorter plateau vanishes [Fig. 8(a), with a pulse width 0.044T]. The duty cycle asymmetry also increases moving the offset phases away from Φ₁ +Φ₂ = −π/2. The minimum duty cycle is set by the transition time between the square-wave plateaus, which is determined by τ; thus, it is of order ε in dimensionless time. Other than that, the duty cycle can be conveniently tuned over the whole period, while keeping the period constant, by changing the pump or the DC voltages applied to the MZIs as shown in Ref. [29] for Φ₁ = Φ₂.

 

Fig. 7 Fundamental out-of-phase solution for Φ₁ = −0.1π, Φ₂ = −0.4π, Φ₀ = 0, γ_ii = 0.5, T_f = 40ns, T_c = 60ns, ε = 4.17 × 10⁻⁴, δ = 1.2 × 10⁻², P = 1.702 (a), P = 1.72 (b), and P = 1.8 (c). P_c = 1.7013.

Download Full Size | PDF

 

Fig. 8 Fundamental out-of-phase solution for Φ₁ = −0.25π, Φ₂ = −0.15π, T_f = 40ns, T_c = 60ns, ε = 4.17×10⁻⁴, δ = 1.2×10⁻², P = 1.616 (a), P = 1.6271 (just below threshold) (b), P = 1.7 (c), and P = 2 (d). Other parameters as in Fig. 5.

Download Full Size | PDF

We now focus on the microsecond oscillations associated to j = 0 (k′ = k = 0). Neglecting ε and δ, according to Eq. (11), ω = 0. For finite ε and δ, ω is slightly shifted from Eq. (11):

Therefore ω₀ = α₀ ∝ δ^1/2 in the MHz regime typically. As shown in Fig. 6 these solutions have a threshold value which is practically independent of s₀ (green line) and which takes place at a value slightly larger than the minimum threshold for square waves [29]. Low frequency Hopf instabilities are also found in a single OEO with positive feedback [23, 40] and can be used for pure microwave generation [21,41,42]. For two coupled OEOs Fig. 9 shows the shape of a microsecond solution with asymmetric duty cycle for different pump values. Increasing the pump the asymmetry of the duty cycle increases and the solution shows a tilted plateau, however it never takes a full square wave shape, rather becomes chaotic. The period of these solutions increases with the pump [29] as clearly seen in the figure. For identical γ_ij and offset phases, microsecond solutions are in phase and have exactly the same amplitude. When offset phases differ, as in Fig. 9, they have a small dephasing as shown in panel (d). The dephasing is so small that at the scale of the other panels the traces for x₁ and x₂ overlap.

 

Fig. 9 Microsecond solution obtained for Φ₁ = −0.15π, Φ₂ = −0.1π, and P = 1.444768 (a), P = 1.4538 (b), and P = 1.454 (c). Panel (d) shows a zoom of c) close to the maximum. Other parameters as in Fig. 7. The threshold is P_c = 1.4392 and at threshold ω₀ = 0.1215. Time traces for x₁ and x₂ overlap in panels (a)–(c).

Download Full Size | PDF

6. Onset of T/4 dephased periodic square waves

Here we consider the parameter regions for which the instabilities of the steady state associated to k odd dominate. As shown in Figs. 2 and 3, in the (Φ₁, Φ₂) parameter space, this typicality occurs in the second and fourth quadrant where offset phases have opposite signs.

To illustrate the scenario, let us consider γ_ij = 0.5, Φ₁ = −0.25π and Φ₂ = 0.15π, parameters for which increasing the pump the first instabilities are those associated to k odd and k′ even (P_c = 1.1188) [Fig. 2(d)] followed by those with k and k′ odd at P_c = 1.2526 [Fig. 2(c)]. Instabilities with k even are not relevant (either are not allowed or have a very large threshold [Figs. 2(a), 2(b)]). Figure 10 shows the Hopf bifurcations in the range ω < 8π and 0 < P_c < 1.8. In addition to P_c and ω obtained solving Eqs. (9) and (10) numerically, we have plotted the phase difference φ₂ − φ₁ from Eq. (20). Families of Hopf bifurcation lines are born at rational values of the delay ratio s₀ = q′/q, irreducible. The members of a family can be identified by j, such that 2k′ = jq′ and k = jq. Since k is odd, q and j must be odd; and since 2k′ is even, q′ must be even. Thus Hopf lines are born at even-odd rational values for s₀ and only odd harmonics are allowed. The stability analysis also predicts that instabilities arising at q′ doubly even (k′ even) have a lower threshold than those at q′ singly even (k′ odd) as shown in Fig. 10(a). The amplitude ratio and dephasing between x₁ and x₂ are given by the amplitude and phase of Q. At the onset of the Hopf lines, from Eq. (18)Q is purely imaginary, so there are two kinds of solutions: those where φ₂ −φ₁ = π/2 for which x₂ is delayed a quarter of the period with respect to x₁ (dephasing T/4), and those where φ₂ −φ₁ = −π/2 for which x₂ advances x₁ by a quarter of the period (dephasing −T/4). For the parameters considered in Fig. 10, |Q| = 0.448 for q′ doubly even while |Q| = 2.621 for q′ singly even. In both cases [1 + P_cF₁ (−1)^k′]/(P_cK₁) > 0, thus the phase of Q is π/2 for k (mod 4) = 3 and −π/2 for k (mod 4) = 1. As a consequence, fundamental solutions born at s₀ with q = 3,7,11,… are dephased T/4 while those born at s₀ with q = 1,5,9,… are dephased −T/4, and successive harmonics born at a given s₀ have alternatively positive and negative detunings as shown in Fig. 10. According to Eq. (20), changing s₀ with respect to the rational q′/q leads to a change in the dephasing as shown in Fig. 10.

 

Fig. 10 Hopf lines for Φ₁ = −0.25π and Φ₂ = 0.15π. Other parameters as in Fig. 4. Yellow and blue lines correspond to bifurcations leading to synchronized solutions dephased +T/4 and −T/4, respectively.

Download Full Size | PDF

T/4 square waves have a symmetric duty cycle and multiple stable harmonics can coexist, as illustrated in Fig. 11 for s₀ = 4/3 (q′ doubly even with lower threshold) and for s₀ = 2/3 (q′ singly even with larger P_c). In agreement with the stability analysis, for s₀ = 4/3 the amplitude of x₂ is smaller than that of x₁ while the opposite occurs for s₀ = 2/3. Also, while P is the same, owing to the lower threshold, the overall amplitude of the square waves for s₀ = 4/3 is larger. Nevertheless, in both cases the fundamental solution corresponds to k = 3, and according to Eq. (14) has a period T = 4/3 (80ns since T_c = 60ns). First and second harmonics correspond respectively to j = 3 (k = 9) and j = 5 (k = 15) with period T = 4/9 (26.6ns) and 4/15 (16ns). For the fundamental and second harmonic x₂ is delayed T/4 with respect to x₁, while for the first x₁ is delayed T/4 with respect to x₂ (dephasing −T/4). Higher order harmonics are also found provided the duration of the plateau is longer than the transition time which, as discussed before, depends on τ. Figures 11(d) and (h) show the 30^th harmonic, j = 61 (k = 183) with a period T = 4/183 (1.31ns). For τ = 25ps as used in Fig. 11 harmonics above 40 are unstable. However, for smaller τ, one can generate T/4 synchronized square waves with a subnanosecond period as shown in Fig. 12.

 

Fig. 11 Coexistence of T/4 square waves for the parameters of Fig. 10, P = 1.3, and T_c = 60ns. Fundamental, 1^st, 2^nd and 30^th harmonics for T_f = 80ns (s₀ = 4/3) are shown in (a)–(d) while (e)–(f) show the corresponding ones for T_f = 40ns (s₀ = 2/3). Black and red lines correspond to x₁ and x₂ respectively.

Download Full Size | PDF

 

Fig. 12 Higher harmonics for the parameters of Fig. 11 but τ = 10ns and s₀ = 4/3: (a) 40^th, T = 4/243 (0.99ns), (b) 60^th, T = 4/363 (0.66ns) and (c) 80^th, T = 4/483 (0.50ns).

Download Full Size | PDF

T/4 oscillatory solutions are born supercritically with a sinusoidal shape as shown in Fig. 13(a) for the fundamental solution born at s₀ = 4/3. Increasing the pump, the solution soon becomes square-wave shaped as it can be seen in Figs. 13(b) and Figs. 13(c).

 

Fig. 13 Square-waves dephased T/4 for T_c = 60ns, T_f = 80ns, P = 1.119 (a), P = 1.13 (b), and P = 1.25 (c). Other parameters as in Fig. 11.

Download Full Size | PDF

7. Robustness of T/4 dephased square waves to delay time mismatch

We have shown that the Hopf lines leading to different kinds of synchronized square waves originate at suitable rational values for the ratio between the delay times, thus from an practical point of view, it is important to assess the robustness of the synchronized square waves to small mismatches in the delay times with respect to the ideal ratio. For in- and out-of-phase square waves arising for identical offset phases, we have shown that while synchronization is robust to mismatches up to a 5%, solutions with asymmetric duty cycle born for positive feedback are more robust than those with symmetric duty cycle arising for negative feedback, and that the larger the period of the solution the higher the robustness of the synchronization [28,29]. In this section we focus on the effect of a small mismatch in the delay times on the period, the shape and the synchronization of the T/4 dephased square waves.

As an example, we consider the fundamental and the first harmonic square-waves originated from the Hopf lines that in Fig. 10 exhibit a minimum at s₀ = 4/3 (plotted in yellow and blue respectively). The left panels on Figs. 14 and 15 show the time traces obtained integrating numerically Eq. (3) for P = 1.25, keeping fixed T_c and changing T_f. Figure 14 has been obtained using the fundamental square wave (T = 4/3) as initial condition while the first harmonic (T = 4/9) has been used for Fig. 15. With 1% mismatch [Figs. 14(b) and Figs. 15(c)] the solutions exhibit longer transition times between the two the plateaus while the plateau lengths are reduced. Furthermore the slope of the positive jump (transition from the negative to the positive plateau) is different than the slope of the negative jump (transition from the positive to the negative plateau). For a positive mismatch with respect to the ideal ratio, as in Fig. 14, the duration of the positive jump is larger than that of the negative jump, whereas for a negative mismatch is the other way around. Increasing the mismatch, the duration of the jumps becomes larger while the plateaus shorten and the difference between the positive and negative jumps becomes more apparent. Through this mechanism T/4 have a large degree of flexibility making them more robust to delay-time mismatches than in- and out-of-phase square waves with symmetric duty cycle arising for negative feedback where already for 2% mismatch the transition jumps develop intermediate plateaus [28]. Note that since self- and cross-coupling delays correspond to fiber lengths of several meters mismatches below 1% are easily achievable in practice.

 

Fig. 14 (a)–(d): Robustness of the T/4 dephased fundamental solution changing T_f : 80ns (s₀ = 4/3) (a), 81ns (s₀ = 1.350) (b), 82ns (s₀ = 1.367) (c), and 83ns (s₀ = 1.383) (d). P = 1.25, other parameters as in Fig. 11. We also show the frequency of x₁ (e) and the phase difference between x₂ and x₁ (f) as a function of s₀. Red dots correspond to numerical simulations of (3) and solid black lines to the theoretical prediction from (9) and (10).

Download Full Size | PDF

 

Fig. 15 As Fig. 14 for the first harmonic with T_f = 80ns (s₀ = 4/3) (a), T_f = 80.5ns (s₀ = 1.342) (b), T_f = 81ns (s₀ = 1.350) (c) and T_f = 81.5ns (s₀ = 1.358) (d).

Download Full Size | PDF

Figures 14(e) and Figures 14(f) show the frequency of x₁ and the phase difference between x₂ and x₁ as a function of s₀ for the fundamental square wave. The solid line corresponds to the prediction of Eq. (20) whereas red dots have been obtained integrating Eq. (3) numerically and applying the Poincaré section technique. The synchronized fundamental square wave has a practically constant frequency up to 5% mismatch, value at which it becomes unstable and the system jumps to a higher frequency square wave. Within this range the phase difference changes in a practically linear way as predicted theoretically. Higher harmonics are more sensible to mismatches in the delay times. For instance, as shown in Figs. 15(e) and Figs. 15(f), the first harmonic becomes unstable with 2% mismatch. This is in agreement with the linear stability analysis as it can be seen from Fig. 10 where the Hopf bifurcation curves for higher the frequency solutions are sharper. From Fig. 10, we can also argue the reason why the shape of the square waves smoothen: increasing the mismatch while keeping fixed the pump P leads to an increase in P_c along the Hopf branch, leading to a reduction of the distance P − P_c to the Hopf bifurcation where the solution is sinusoidal. For larger values of P the solutions are less robust to delay-time mismatches and even become chaotic.

8. Final discussion and remarks

We have explored the synchronized square wave solutions generated by two delay-coupled OEOs as function of the offset phases and the ratio between self- and cross-delay times s₀. Despite the OEOs being different, synchronized square waves originate from Hopf instabilities of the steady state and multiple harmonics coexist. For both OEO with negative feedback square waves have a symmetric duty cycle and arise supercritically, being synchronized in phase for s₀ being an odd/odd rational and out of phase for s₀ odd/even rational. For both OEO with positive feedback square waves have an asymmetric duty cycle and are born subcritically. In this case for s₀ rational with an odd numerator square waves are in phase while if the numerator is even both in- and out-of-phase square waves coexist. Increasing the pump, subcritical asymmetric square waves are also found beyond the parameter regions where they are born, including regions of negative feedback [29].

We have also shown that for mixed feedback the OEOs show an interesting dynamical regime in which synchronized square waves dephased by a quarter of the period are stable. Furthermore we have shown that this type of synchronization, which does not arise for identical OEOs, exists for a broad range of parameter values and is robust to small mismatches in the delay times.

Altogether this system displays a very large degree of flexibility allowing for instance to select the period of the square waves without changing parameters by choosing a suitable initial condition, to tune the duty cycle while keeping the period constant just changing the offset phase or the pump level and to tune the dephasing between the synchronized square waves by changing the offset phases or the delay times. The methodology and results presented here can be generalized to other systems composed by several oscillators subject to mixed feedback.