Optical Parametric Oscillation with Ultra-Low Power Threshold in a Dimer of Active-Passive Cavities

: Optical parametric oscillation can convert the input laser into a couple of coherent optical output with signal and idler frequencies. It is an important method for the realization of the broadband middle infrared tunable lasers. The optical parametric oscillation process depends on the ultra-fast non-linear response of matter to light with a certain pump power. Therefore, reducing the pump threshold of the optical parametric oscillation process and improving the energy conversion efﬁciency are of great signiﬁcance to the study of non-linear optics. In this paper, we construct a dimer system that couples a passive non-linear resonator to an active resonator. Based on the dimer system, an ultra-low threshold optical parametric oscillation generation method is proposed. By coupling the gain pump mode, the non-linear effect is effectively enhanced, the pump power threshold is reduced to half of when there is no gain, and the energy conversion efﬁciency is increased. We believe this research provides a feasible method for a low-threshold tunable and easy-to-integrate optical parametric oscillation laser source.


Introduction
The optical parametric oscillator converts the input pump laser light into two output lights with lower frequency (usually called signal light and idler light) through the secondorder nonlinear optical interaction. Specifically, the optical parametric oscillations (OPO) describes the non-linear process that occurs when the pump light passes through the χ (2) material in the optical resonator, which converts a pump photon with frequency ω p into a signal photon with frequency ω s and an idler photon with frequency ω i under the phase matching conditions [1][2][3][4][5][6]. In the field of quantum optics and quantum information science, the OPOs are often used as the coherent light source to generate the optical squeezed states [7], entangled photon pairs [8][9][10][11][12], or the correlated optical network [13].
Similarly to other non-linear optical effects, the OPOs process also has a power threshold due to the optical loss and modal phase matching condition of the materials [14]. For example, OPO thresholds have been observed in bulk lithium niobate whispering gallery resonators with 6 µW [15], milliwatt-level OPOs thresholds have been observed in the planar periodically polarized Ti:LiNbO 3 waveguide [16]. Additionally, the thresholds of OPOs were observed in GaAs [17] and GaAs/AlGaAs [18] ridge waveguides are of 5.7 W and 210 mW, respectively. In order to improve the performance of the OPOs, the method by reducing the optical loss of the material is always considered as an important and efficient way. Relying on the specific optical device with low optical loss and field enhancement, such as the optical microcavities, the threshold of the optical non-linear effects could be improved [19,20]. Here in this study, we intend to propose a novel platform based on the whispering gallery mode (WGM) optical microcavities to show the better performance of OPOs. The WGM optical microcavity describes the optical dielectric resonator under the rotational symmetry which is the optical analogue of whispering gallery in acoustics [21].
Due to the special structure, the optical field could be totally reflected by the inner surface and the WGM microcavity usually shows a high quality factor, as the fields are resonantly enhanced in it. Recently, the studies of optical microcavities with ultra-high quality factor have been made remarkable progress in the field of light-matter interaction [22,23]. Due to the ultra-high quality factor and a small mode volume, it can greatly enhance the nonlinear interaction. During the past few decades, various studies have been made based on WGM microcavities in the field of non-linear effects, such as the second harmonics [24][25][26], the OPOs [15,27], the photon blockade [28][29][30], and optical frequency combs [31,32].
In addition, the photonic molecule structure could be realized by coupling the two WGM microcavities [33]. Moreover, if we choose an active microcavity with optical gain provided to a dissipative microcavity, by properly choosing the optical gain and loss, the optical dimer with parity-time (PT) symmetry could be obtained [34,35]. Employing the parity-time symmetry, the low threshold PT-symmetric lasers could be realized [36][37][38][39][40]. Additionally, in the field of non-linear optics, the low optical loss may significantly enhance the non-linear effect and improve the performances of high-order sideband generation [41] and the second harmonic generation [42]. As the field distribution in the PT-symmetric structure is sensitive to the environment, it can also be used as the ultra-sensitive detector for nanoparticles [43][44][45][46].
Here in this paper, we propose a dimer system in which a passive WGM resonator made by the material with second-order non-linear coefficient χ (2) is coupled with an active resonator with optical gains. The system works in the OPOs regime by a visible light pumping to produce infrared signal and idler photon pairs. The OPOs process is theoretically studied, the threshold power and the spectrum are both obtained numerically. We find that by tuning the coupling strength between the resonators, due to the energy exchange between the gain cavity and the dissipative cavity, the pump power threshold can be effectively reduced and the energy conversion efficiency can be increased. Compared with previous results, our research has greatly improved the non-linear conversion efficiency.

The Theoretical Model
The model for optical parametric oscillation generation is a dimer configuration of coupled passive-active microresonators, as shown in Figure 1. The second-order nonlinearity in the optical passive (loss) resonator is denoted as χ (2) , a visible pump laser (represented byâ) at an angular frequency ω a pumps the microresonator. We consider the degenerate OPOs process which produces a degenerate signal photon pair (represented bŷ b) with frequency ω b = ω a /2, and this process satisfies the energy-conservation and phase matching condition. Here, the coupling strength between the pump photon and signal photon under the second-order non-linearity is denoted by g. Meanwhile, an additional single-mode active micro-resonator (denoted as modeĉ, with frequency ω c ω a ) is used to evanescently coupled to the pump modeâ of the passive micro-resonator via their spectral mode overlap, with the coupling strength J.
The system with the pump laser driving under the frequency ω p ω a can be described by the Hamiltonian:Ĥ Here,Ĥ 0 is the energy of the total three cavity modes without considering the coupling between modes, it can be expressed aŝ whereâ † (â),b † (b), andĉ † (ĉ) are the bosonic creation (annihilation) operators of the pump, signal, and active optical modes, respectively. Schematic of a dimer with two active-passive micro-resonators to enhance the OPOs process. An optical passive (loss) micro-resonator with the second-order non-linearity χ (2) (the red one), which has an input pump mode (frequency ω a ) and a signal mode (frequency ω b ω a /2) generated by the pump laser through the optical parametric oscillation process. Meanwhile, the pump laser in the passive micro-resonator is coupled into the active micro-resonator (the blue one) with coupling strength J. The pump laser (ω p ω a ) drives the passive micro-resonator through the two fiber tapers.
AndĤ ab represents the interaction between the pump photon and the signal photons in the lossy micro-resonator which can be expressed aŝ where g is the coupling strength between modeâ andb due to the non-linearity χ (2) which can be expressed as: where n e f f is the refractive index of the phase-matched pump and signal modes, χ (2) is the intensity of second-order non-linear coefficient of the materials, ε 0 is vacuum permittivity, and R is the radius of the passive micro-resonator. ζ denotes the effective mode overlap factor on the cross-section of the micro-resonator. The third itemĤ ac describes the mutual coupling between the pump photons in the passive micro-resonator and the single mode photons of the active micro-resonator, in the formĤ where J represents the coupling strength which can be controlled by tuning the distance between the two micro-resonators. Finally, the last itemĤ d =h √ κ a,1 (ε pâ † e −iω p t + ε * pâ e iω p t ) describes the external driving field, in which the pump frequency ω p , the pump amplitude ε p , and the pump power P p satisfy the relationship ε p = P p /(hω a ). The loss of the micro-resonator can be described by κ a(b,c) = κ (a(b,c),0) + κ (a(b,c),1) , κ (a(b,c),0) represents the inherent dissipation of the microresonator and κ (a(b,c),1) represents the coupling dissipation between the micro-resonator and the fiber taper, in which κ (a(b),0) > 0 for the lossy cavity and κ (c,0) < 0 for the gain cavity. By applying the unitary evolution operationÛ(t) = exp[−iω p (2â †â +b †b +ĉ †ĉ )t] and regardingh as 1, the total Hamiltonian of the system becomeŝ where δ a = ω a − ω p , δ b = ω b − 2ω p , and δ c = ω c − ω p represents the frequency detunings of the input modes, signal modes, and active cavity modes from the pump laser, respectively. By considering the addition of dissipation and neglecting the quantum noise term, the dynamics of the bosonic operators can be expressed as: where κ a , κ b , and κ c are the decay rates of the input modes, signal modes, and active cavity modes, respectively. Under the non-depletion and omit the weaker signal modes effects terms, we solve the equations under the steady-state approximations as Then, the dynamics of b can be expressed as where b in describes the vacuum noise. By performing the Fourier transformation, the operators in time domain can be transformed into the frequency domain as As the spectrum density around mode b satisfies S b (ω, ω ) = 2κ b,1 b † (ω)b(ω ) , we finally obtain: We choose the condition where the input pump power falls below the threshold of the OPOs: (δ 2 b + κ 2 b ) 4g 2 α. By neglecting the 4g 2 α term in the denominator, the signal spectrum density S b (ω) becomes and the total generation rate of the signal photons could be expressed as: The pump threshold for OPOs satisfies the relation that the signal photon generation rate is larger than the loss of the signal mode, so the pump threshold could be expressed as When the system meets the resonant condition as (δ a(b,c) = 0), we have Meanwhile, according to Equations (8) and (9), we can obtain By substituting the Equations (21) and (22) into Equation (9), under resonance conditions (δ a(b,c) = 0) we can get If we define the single-photon cooperativity as Then, Equation (23) can be re-written as Therefore, the power of the signal photons could be solved as:

Results and Discussion for Enhanced Opos
OPOs has been experimentally studied in various materials, such as in periodically polarized lithium niobate waveguide (PPLN) [15], aluminum nitride (AlN) microcavities [27], and photonic crystals. For example, in Reference [27], a dual-resonant phase-matched AlN micro-ring resonator is used to demonstrate low-threshold parametric oscillations in the telecom infrared band, where the χ (2) non-linearity and the coupling strength (g) of dual-band can be precisely controlled by nano-manufacturing technology.
According to the results in the previous studies, the dissipation of the pump mode is often larger than that of the signal light mode. Therefore, we assist an active microresonator in the system to increase the pump mode in the passive micro-resonator through the optical field coupling, to improve the performance of non-linear effects. Here we choose the parameters in our numerical simulation as, the non-linear coupling strength g produced by the χ (2) non-linearity is g/2π = 80 kHz refers to the parameters obtained from the experiment in Reference [27], the pump laser mode wavelength is λ a = 780 nm, the signal mode wavelength generated by the optical parameter oscillation process is λ b = 1560 nm, the intrinsic optical factors Q 0 of the pump mode and the signal mode are 1.0 × 10 5 and 2.0 × 10 5 , respectively. The total quality factor Q of the pump mode and the signal mode are 2.0 × 10 5 and 4.0 × 10 5 (corresponding to the total loss rate κ a /2π = 1.92 GHz, κ b /2π = 480 MHz, the coupling loss rate κ a,1 /2π = 960 MHz, κ b,1 /2π = 240 MHz).
The material of the passive micro-resonator is AlN [27]. The active micro-resonator can use a thulium ion (or zirconium) doped silica resonator, and can be excited by a pump laser doped with rare earth ions to generate a gain mode, based on a micro-resonator made of Ti : Al 2 O 3 , or a tunable droplet micro-resonator based on electro-wetting [47][48][49][50][51]. Here we set the coupling strength as J/2π = 0.8 GHz, which can be controlled by tuning the gap distance between two micro-resonators [52]. The phase matching between the input mode a and the signal mode b in the micro-resonator can be controlled by the width of the micro-resonator roughly meet the phase matching, and fine-tuned by thermal tuning [27]. Besides, the gain modes of the active micro-resonator is to detune the signal modes of the passive micro-resonator.
To show the influence of the active (gain) micro-resonator on the OPOs process of the system. We first study the relationship between the coupling strength J and the pump power threshold (P th ) of OPO, based on Equation (20), and the result is shown in Figure 2.
Obviously, when J = 0, it is the original system without gain. Mathematically, the Equation (20) will also become the form of the OPOs pump power threshold in Reference [27], and the pump power threshold of the system without gain is about 26.3 mW. After the active cavity is coupled with the coupling strength J = 800 MHz and the gain mode κ c = −1 GHz, the pump power threshold is reduced to about 11.7 mW. We found that the introduction of gain significantly reduced the OPOs threshold. In Figure 2, we find that as the coupling strength (J) increases, the pump power threshold for OPOs decreases significantly. This can be understood as the gain of the active (gain) cavity coupled to the passive (dissipative) cavity compensates for part of the dissipation and enhances the χ (2) non-linear process. When we choose a larger coupling strength J, that is, by reducing the coupling distance between the two micro-resonators, we can reduce the pump power threshold for OPOs generation reduced to half of when there is no gain. By calculation of the signal light power, we found in Equation (26) that in order to generate the signal light more effectively, it is not required to increase the power of the pump source. In order to study the relationship between coupling strength and energy conversion efficiency in the OPOs process, we introduce energy conversion efficiency as We present the relationship between energy efficiency and pump power, which is shown in Figure 3a.
We find that energy conversion efficiency η is related to the pump power P p , the mode dissipation rate (κ a(b,c) , κ a,1(b,1) ), and the coupling strength J. Different from the second harmonic generation process by increasing the pump power to increase the energy conversion efficiency (J = 0, η = 25% corresponding to Figure 3b when there is no coupling), when the pump power 4P = P th , we can obtain the maximal energy conversion efficiency η = 37% which is much higher than the case of no gain cavity. From Equation (27), we find that it is very effective to increase energy conversion efficiency by increasing κ a(b),1 and the coupling strength J. This is of great significance to our modulate system. For example, over-coupling will not only increase the energy conversion efficiency but also increase the pump power threshold. Meanwhile, we can increase the energy conversion efficiency by increasing the coupling strength between the resonators and reduce the pump power threshold. As shown in Figure 3b, we can change the energy efficiency η by tuning the gap distance between the micro-resonators. The reason is that as the coupling strength between the resonators increases, the gain cavity compensates for the loss of mode a of the dissipative cavity. The coupling gain of the pump light field enhances the non-linear process in the dissipative cavity and improves the energy conversion efficiency of the OPOs process.
Here we only consider the case that the system is under the resonant condition in the above, and we did not consider the impact of the detuning between the pump power and modes a, b, and c on the pump power threshold. In the following, we will discuss the impact of detuning on the pump power for the system to produce the OPOs process. Equation (19) is used to calculate the pump power threshold changes in the presence of various detunings, as shown in Figure 4. We find that the pump power threshold increases with the detuning, regardless of the detuning direction. This is because the interaction induced by the second-order nonlinearity is weakened in the non-resonant case, and the non-resonant coupling with the gain cavity prevents the decrease in the optical field intensity in the loss cavity. These will inhibit the generation of the optical parametric oscillation process and increase the pump threshold. Therefore, in order to obtain a lower pump power threshold, it is required to put the system in resonance as much as possible.

Summary
In summary, we proposed a dimer system consists of a passive (dissipative) microresonator with χ (2) non-linearity coupled with an active (gain) micro-resonator, in which the optical parametric oscillation is theoretically studied. By tuning the coupling strength between resonators, the pump power threshold that generates the OPOs process could be reduced and its energy conversion efficiency could be improved. Within the range of adjustable coupling strength, we can reduce the pump power threshold to half of the power without gain and increase its energy conversion efficiency to 37%. This work has adopted feasible technologies which greatly improve the performance of the single resonator to generate the OPOs, and it also provides a feasible way for the development of high efficiency and low threshold tunable OPOs laser source.