Synchronization of Two Indirectly Coupled Singly Resonant Optical Parametric Oscillators

We analyse a system of a singly resonant optical parametric oscillator for a second order nonlinear material. First, we show that the dynamics of the resonating cavity signal mode can be expressed by a Stuart-Landau oscillator, for a certain pumping powers close to the threshold. Second, we couple two optical parametric oscillators indirectly via a cold resonator. When the condition of a weak coupling is satisﬁed, the limit-cycle of each oscillator is unaltered, and the sys-tem is described by a coupled phase oscillator model (Kuramoto model), where a frequency synchronization of the two oscillators occurs at a critical coupling constant.


Introduction
When a laser light pumps a second order nonlinear material in a resonant cavity, a down conversion processes occur at a certain threshold pumping power.Thus, a pump photon of a frequency ω p generates two photons of frequencies ω s and ω i .These two photons ω s and ω i are usually called signal and idler, respectively.As light from the down conversion process is contained in a resonant optical cavity, the nonlinear gain is enhanced, and optical parametric oscillations are sustained [1,2].Thus, we recognize the behavior of the optical parametric oscillators (OPOs) appearing due to a conservation of the number of photons inside the resonator.Systems of OPOs represent robust sources of coherent light and tunable radiation.In addition, OPOs are efficient sources of non-classical light states such as bi-partite entangled states, optical twin beams and squeezed vacuum states [3][4][5][6][7][8].
Singly resonant optical parametric oscillators (SROs) are obtained once the cavity resonates at a signal frequency [8][9][10][11][12][13].A qualitative perspective that stimulates the comprehension of SROs features is provided by investigation of the classical characteristics of light emitted from OPOs [14,15].The SROs offer single frequency light sources with a narrow linewidth of the output radiation, making these devices highly recommended for a wide area of applications such as remote sensing, spectroscopy analysis, photoelectric detection, and optical frequency division [3][4][5][6][7][8][9][10][11][12][13].In previous works, the self-sustained oscillations of the field in OPOs devices are treated as a limit-cycle oscillation but the type of the limit-cycle is not investigated [14,16,17].
In this work, attention is given to the theoretical investigation of the self-sustained oscillations of the SRO?s output field.Our findings provide a description of the SRO?s signal field oscillation by using the Stuart-Landau oscillator model marginally exceeding a threshold pump amplitude.Thence, we introduce the limit-cycle solution of the Stuart-Landau model showing that this solution is equivalent to the solutions of classical coupled mode equations [18,19].As a result, we couple two SROs of slightly detuned frequencies, indirectly by incorporating a cold cavity between them.This coupling technique avoids any time delay and strong coupling effects such as mode competition and unstable perturbation of the SROs [20].We demonstrate that the system is modeled by two Stuart-Landau oscillators coupled to each other through the cold cavity (treated as a harmonic oscillator) and synchronize their frequencies to a common value at a coupling strength threshold.This frequency synchronization indicates that the interacting fields are locked in phase.Our results prove that the amplitudes of the signal field oscillations become time-independent and similar to each other at any value of the coupling strength.Henceforth, we show that the interaction between the two coupled SROs system is described by the Kuramoto model of coupled phase oscillators.
This work is organized as follows: in Section (2), we present the derivation of Stuart-Landau equation with the relevant parameters in order to describe the limit-cycle behavior of an SRO.In addition, we show that the SRO is described by a Stuart-Landau oscillator model.In Section (3), we investigate the system of two indirectly coupled SROs.We also show in Section (3) how the coupled SROs are described by a coupled phase model (Kuramoto model of two oscillators).We account on the synchronization feature of two SROs in Section (4).A conclusion is provided in the Section (5).

Limit-cycle behavior of an SRO
We present the SRO by considering an optical cavity of resonance frequency ω s containing a second order nonlinear crystal [8,18,19], as shown in Fig. 1a.The cavity is pumped by an external laser at ω p with a pump amplitude F p and assuming, for simplicity that the external pump phase matches the cavity pumped mode.The Hamiltonian describing this system has the structure: H = H o +H pump +H int .The harmonic oscillator Hamiltonian is H o , H pump is the pump Hamiltonian and H int is the nonlinear interaction Hamiltonian that represents the creation of a signal-idler pair by annihilation of one pump photon under a perfect phase matching condition, as seen in Fig. 1b.In a single mode operation, these Hamiltonians can have the expressions (in the limit ℏ = 1) [2,14]: In Eq. ( 1), a j and a † j are the bosonic creation and annihilation operators, for the mode j, where j = p, s, i.The real constant g represents the nonlinear interaction strength, which depends on the second order susceptibility χ 2 [18,19].The pump field is coupled into the cavity at a rate γ p which represents the cavity decay rate for the pump photons.
The classical equations of the amplitudes for the system (1) are obtained by applying the Heisenberg?s equations of motion: ȧj = −i[a j , H] taking into consideration that there is a large number of photons in each mode.Thus, < a j >= α j where the quantities α j represent the (complex) electric field amplitude associated to the mode j for j = p, s, i.This process leads to the following expressions [14,18,19]: where γ s and γ i are the photon decay rates of the signal and idler modes, γ j = ω j /Q j where Q j is the quality factor of the optical resonator for the mode j = p, s, i.The threshold pump amplitude for Eq. ( 2) is given by |F (p,th) | = γ p γ i γ s /16g 2 [18,19].For a singly resonant operation, losses for the pumped and idler modes exceed the losses experienced by the signal mode: γ p γ i ≪ γ − s.Henceforward, the pump and idler waves reach a steady state much faster than the signal wave [18,19], and their time evolution reaches zero.While the self-sustained oscillation of the SRO?s is considered as a limit-cycle oscillation, the category of this limit-cycle is not known.Therefore, we need to show the time evolution of SRO, αs , in a defined (even for the best approximation) form of a limit cycle oscillator.In addition, it is essential to present this limit cycle form in a valid range of the pump above the threshold.In order to achieve this, first we solve system (2) in the steady state to obtain values of α p and α i .Because αp and αi go to zero earlier than αs , we expect that obtaining α i and α i by using Eq. ( 2) in the steady state will not much affect the dynamics of αs .Thus, employ the polar form to every complex amplitude as α j = ρ j exp(−iθ j ) for j = p, s, i, then Eq. ( 2) leads to: where ρp = 0 and ρs = 0. Thenceforward, from Eq. (3b) we find ρ i = (2g/γ i )ρ p ρ s .We use the pump Eq. (3a) to find an expression for the pumped mode amplitude ρ p in terms of ρ s .Therefore, by employing Eq. (3a) and Eq.(3b), we have for any pump parameter ε > 1, the amplitude of the signal mode is expressed as where (|α s |) (eq.2−staedy) is calculated from system (2) at the steady state.The plot in Fig. 2 shows that always the oscillation of the signal is a limit-cycle oscillation for any pump |F p | above the threshold and for any value of the parameter ε.The limit-cycle behavior indicates that the number of photons in the signal mode is sustained and reaches a steady state.This self-sustained oscillation is balanced by the linear gain due to pump, and the nonlinear gain saturation which originates from the parametric down conversion process.However, in order to achieve coupling of signals modes, the expression of the limitcycle oscillator has to be known.Consequently, by using ρ p = √ γ s γ i /2g and ρ i = 2gρ s ρ p /γ i , we may write the amplitude equation Eq. (3c) as ρs = −γ s /rho s /2 + g( √ γ s γ i /2g)(2g/γ i )(ρ s √ γ s γ i /2g) = 0. Thus, we are not able to see how ρs evolves to reach the steady state.Yet, we do not show up the form of the time evolution of ρ s (α s ).Subsequently, to examine a limit-cycle form of the signal mode to obtain an expression of the limit-cycle oscillator, we use the formulas of Under the condition that the external pump amplitude F p is near the threshold, and γ p γ i ≫ g 2 , we can approximate the expansion of |α p | 2 to a first order.Accordingly, the expression 2 can be expanded by the binomial expansion as ε ≳ 1 to have the form: Afterward, by substituting |α i | = 2g|α s ||α p |/γ i and |α p | from the above relation into the evolution equation of the signal mode (the last relation of system ( 2)) , we obtain: where 4g ).We notice that the third and last terms contain , ε is the pump parameter near threshold as ε ≳ 1 [21].Then, after rearrangements we reach to the time evolution of the complex amplitude of the singly resonant signal as Thus, Eq. ( 5) takes the form of the Stuart-Landau oscillator model [21], which is given by: αSL where A = γ s (ε 2 − 1)/2 is the gain term, B = (γ s ε)/2|F p,th | 2 is the gain saturation.
In addition, the signal mode, according to Eq. ( 6), has an amplitude: which represents the amplitude or equivalently the size of the limit-cycle of the oscillation.The limit-cycle behavior of an SRO, according to relations ( 6) and ( 7), is understood by the fact that the oscillation of the signal mode is sustained by the energy supply due to pump, the nonlinear gain saturation due to the down conversion and the cavity losses [18,19].It is necessary to find the effective range of the pump parameter ε, in order to obtain |α SL | ∼ = (|α s |) eq.2−steady .Fig. 3 demonstrates that Eq.( 4) and Eq.(7) coincide for a certain range of the parameter ε.Therefore, for the pump parameter ε is remarkably close to unity, the ratio (|α s | 2 ) eq.2−steady /|α SL | 2 = 2ε 2 /(1 + ε) ≃ 1 and the two models (system (2) and Eq. ( 6) are equivalent.So, we pump the resonator close to the threshold and investigate the temporal evolution of every amplitude of the pump, the signal and the idler in order to get the time evolution of α s .Hence, the Stuart-Landau model completely agrees with the coupled mode model in describing the evolution of the SRO near threshold pumping and provides a further insight in describing the limit-cycle behavior of the SRO that can be employed into the modeling of coupled SROs as coupled limit-cycles.

Indirectly coupled optical parametric oscillators
In this section, we introduce the system of two indirectly coupled SROs, as shown in Fig. 4. The two SROs have similar photon decay rates and a nonlinear coupling parameter.Both two cavities 1 and 2 are pumped slightly larger than the threshold.The main purposes behind coupling between the two SROs are to achieve frequency synchronization of the two lights come from the two cavities 1 and 2 as well as to achieve at least an output intensity twice that of each individual SRO.The only difference between the SROs is their signal frequencies (ω 1 , ω 2 ), where we slightly detune them by an exceedingly small mismatch ∆ = (ω 1 − ω 2 = 1 − 0.99 = 0.01 > 0).The Hamiltonian of the system has the structure: is the cold cavity free Hamiltonian, which is expressed in terms of the bosonic operators (a o , a † o ).The two Hamiltonian H 1 as well as H 2 are similar to H in Eq.( 1) for the parametric oscillators ( 1) and ( 2).The coupling Hamiltonian H c is given by: In the coupling Hamiltonian ( 5), the operators a j and a † j are the bosonic annihilation and creation operators for the signal modes in the parametric oscillator j, where j = 1, 2. The constant k is the coupling strength which is related to the rate of photon exchange between the SROs 1, 2 and the cold cavity, as seen by the arrows in Fig. 3.The cold cavity has a resonance frequency of ω o = (ω 1 + ω 2 )/2, and a photon decay rate of Γ.
The classical equations of motion for the coupled system are obtained by the same method used in the preceding section as (we ignore the pump and idler modes): where α 2 and α 2 are complex fields of the two nonlinear cavities 1 and 2. The cold cavity has a complex field α o .The corresponding Stuart-Landau equations are: We use the polar form for each complex amplitude in Eq. ( 9) and Eq. ( 10 The cold cavity amplitude evolution is: Also, the Stuart-Landau relations in (10) become: The corresponding phase evolution for the systems ( 9) and ( 10) is: where expressions in ( 14) are obtained from the coupled mode relations ( 9) and from the Stuart-Landau model (10).The cold cavity phase?s time evolution is: Eqs. (11)(12)(13)(14)(15) show coupling terms through harmonic functions of phases θ 1 , θ 2 and θ o , where the phases θ 1 and θ 2 are coupled indirectly through the phase θ o .Because the main goal is to attain a frequency synchronization, where θ1 = θ2 = ω o having a certain phase lock between θ 1 and θ 2 , at a critical coupling value k c .Thus, in coupled phase models we expect that the time evolution of both phases and amplitudes as well as the phase differences are time independent at synchronization.Therefore, we anticipate, according to Eq. ( 14), that the two harmonic functions possess two opposite values cos(θ o − θ 1 ) = cos(θ o − θ 2 ) in order to reach to θ1 = θ2 = ω o at k c .Also, the phase differences |θ o −θ 1 | and |θ o −θ 2 | become constants as well as they are time independent.Qualitatively, at synchronization, we guess that |θ 1 − θ 2 | = π/2 [22].In addition, the quantities ρ1 , ρ2 , ρo , ρSL1 , and ρSL2 are zeros.The time evolutions of the amplitudes become zeros because at a certain pump energy, the amplitude is constant.Thus the dynamics of the coupled system depend on phases only.In order to investigate how the amplitudes and phases evolve in time, we solve numerically equations (11)(12)(13)(14)(15), and we plot the important results in Fig. 5.The numerical results, as shown in Fig. 5, show that the amplitudes of the SROs are similar at k c (Fig. 5a).At the critical coupling, a frequency synchronization occurs at ω o , where ω o is the mean frequency of the signal modes from the two OPOs (Fig. 5b).Moreover, the frequencies trajectories (< θ1 >) and (< θ2 >) approach symmetrically towards a synchronization at ω o (Fig. 5b).Thus, the average values of phases and amplitudes at the critical coupling strength will be equal to their real-time values as seen in Fig. 5c.Accordingly, at k c the frequencies θ1 = θ2 = ω o .Then we write Eq. ( 14) as ( Eq. ( 14) as from ( 10 -0.05 Eq. ( 14) as from (9) Eq. ( 14) as from (10) Eq. ( 11 ( . By dividing this result we get Thus, we obtain: So, we have an analytic condition for ρ 1 = ρ 2 at k c , which comes clear by using Eq. ( 14) that leads to , where cos(θ o − θ 1 ) < 0. Consequently, squaring both cosines and subtract from 1 as: . However, Eq. ( 12) guides to the expression for the cold cavity amplitude as So, use the cold cavity amplitude equation ρ o Eq. ( 18) with ρ 1 = ρ 2 , we find sin(θ o − θ 1 ) = sin(θ o − θ 1 ).By substituting Eq. ( 15) into Eq.( 13), using the property cos(θ o − sustained oscillation of the signal photons is described by the Stuart-Landau model, which possesses a limit-cycle in an energy range slightly greater than the threshold value of the pump.This limit cycle behavior results from the balance between the linear gain due to pump, the cavity losses and the nonlinear gain saturation above threshold.The analytical and numerical solutions of the Stuart-Landau model and the original coupled mode equations [18,19] are similar in a certain range of the pump parameter ε close to 1 [21].We make use of the limit-cycle modeling of the SRO and apply the method of indirect coupling between two singly resonant OPOs.We find that the interaction between the two resonators is described by the dynamics of the coupled oscillators that can be reduced to the Kuramoto model coupled phase model.The coupling functions are harmonic even functions and contribute equally, but having opposite values, to the amplitudes of the two oscillators to preserve the limit-cycles.Thus, the time evolution of the amplitudes is zero and the amplitudes are constants.Therefore, the dynamics of the coupled system is described by phase evolutions.In Kuramoto coupled phase model of two oscillators, the contributions to the instantaneous frequencies of the two SROs are coming due to odd harmonic functions opposite in sign in order to merge the two frequencies into a synchronized one.
The current work can be expanded in the future to include coupled optical parametric oscillators involving more than two interacting systems in several configurations.These suggested configurations can take locally coupled in a ring [23], locally coupled chains [25] and globally coupled oscillators [24].Linking the present work with the understanding of the solutions of the corresponding Kuramoto oscillators for these coupling configurations is beneficial.The solutions of the Kuramoto model, for instance, are known and can be connected to the systems of coupled SROs for three and four oscillators with local coupling in a ring and global coupling [23,24].However, as there is a solution for this coupling configuration, it is also necessary to investigate any number N of optical parametric oscillators coupled indirectly in a chain, where solutions at synchronization is known [25].

Fig. 1
Fig.1(a) A schematic illustration of a nonlinear frequency conversion in an optical parametric oscillator: one pump photon of frequency ωp is converted to two photons of frequencies ωs (signal) and ω i (idler).The signal photons match the resonant frequency of the resonator.(b) Energy level representation of the three-wave mixing by the second order nonlinear frequency conversion.

Fig. 4 A
Fig. 4 A schematic illustration of the two coupled SROs by an indirect technique.

Fig. 5
Fig. 5 Bifurcation diagrams for two coupled SROs.(a) the effect of coupling on the amplitudes and their time derivatives.(b) The mean values of the instantaneous frequencies of the two SROs and the cold cavity, obtained by varying the coupling strength starting from zero to k > kc.(c) The time evolution of the amplitudes and their time derivatives.The inset of (c) shows the time evolution of the instantaneous frequencies.