Analytic Theory for Parametric Amplification in High-Q Micro-Ring Resonators

Optical parametric amplifiers (OPAs) are optical amplifiers based on four-wave mixing processes, showing great potential for applications in communications, optical signal processing, quantum optics, etc. In recent years, significant progress has been made to integrated OPAs based on highly nonlinear micro-ring resonators (MRRs), benefiting from the greatly enhanced optical-matter interactions. Notable parametric gain becomes available at unprecedented low power levels, allowing for example the source-integrated coherent optical frequency combs. Therefore, an analytical formula for OPAs in high-quality (Q) MRRs is of great importance in the design of optical parametric devices. Analytical theory for OPAs in high-Q MRRs in the high pump power scenario remains elusive, where intensity-dependent nonlinear phase that brings significant parametric gains cannot be ignored. In this work, analytical formulas for the parametric gain and conversion efficiency (CE) of a high-Q MRR with high-pump excitation are derived. We show the interplay between parametric gain and field enhancement: the field enhancement of the signal and idler wave can be greatly boosted due to the compensation of the round-trip loss by the parametric gain, which in turn leads to increased field enhancement as well as greater parametric gain and CE. Our theory agrees well with numerical and experimental results.

processes, originating from the third-order susceptibility (χ (3) ) of the nonlinear materials. OPAs have been recognized as promising candidates in high-speed optical communication systems owing to their extraordinary properties, such as arbitrary center wavelength, large gain bandwidth, and low noise figure [1]. Apart from being solely amplifiers, OPAs have been explored to realize wavelength conversion, phase conjugation, as well as photon-pair generation, facilitating areas from nonlinear optical signal processing (NOSP) [2], [3] to quantum optics [4], [5]. With the rapid development of integrated manufacturing processes, integrated OPAs have been widely investigated in recent years [6], [7], [8], [9]. Among various integrated structures, high-quality(Q) micro-ring resonators (MRRs) show great potential due to enhanced light-matter interaction. On-chip dissipative Kerr soliton combs [10] are observed at unprecedented low power levels, indicating significantly boosted parametric gain in high-Q MRRs. It is thus important to derive analytical formulas for the parametric gain and conversion efficiency (CE) in OPAs, defined as the power of the output signal and idler wave divided by the power of the input signal light, respectively, based on high-Q MRRs. Theories for OPAs in optical fibers have been well developed and documented back in the 2000s [11]. In 2000, Absil et al. derived a simple expression for the CE of MRRs in the low-pump power regime, where the power of the generated idler light is much lower than that of the input signal [12]. Due to greatly enhanced field intensity inside high-Q MRRs, the contributions from the nonlinear phase mismatch that arises from Kerr-induced nonlinear phase shift cannot be neglected. As a result, the physical behavior of parametric processes in high-Q MRRs under high-power excitation is expected to deviate from the low-pump regime. Recently, the analytic theory of parametric processes in lossy integrated waveguides has been developed based on a parametric transfer matrix [13]. In this Letter, we derive an analytical formula for the parametric gain and CE of a high-Q MRR following the same procedure described in [13]. Especially, the interplay between parametric gain and field enhancement for a high-Q MRR under high-power excitation is revealed explicitly for the first time, bring obvious "M" type curve of parametric gain and CE with greatest gain obtained at phase-matched wavelength, showing interesting similarities and differences compared to the results derived under low-pump excitations [12]. Numerical simulations based on a full-map equation [14] and experiments based on a Si 3 N 4 MRR show good consistency with the theoretical prediction.

II. THEORY AND RESULT
We consider an OPA process in an MRR with a radius R based on the degenerate FWM, as sketched in Fig. 1. A pump light and a co-polarized signal light, centered at the frequency ω p and ω s (matching the two resonances of the MRR), respectively, are launched into the bus waveguide and then couple into the microring via evanescent waves. The power coupling coefficient κ and transmission coefficient r of the coupling region satisfy κ 2 + r 2 = 1 under the lossless coupling approximation. The FWM in the MRR is highly efficient thanks to the strong field enhancement (FE), generating the idler wave at the frequency ω i , satisfying 2ω p = ω s + ω i . The FE is defined as the intracavity field divided by the field at the input bus waveguide. Finally, the three waves propagate to the output port after coupling out of the MRR.
We start with a three-wave model describing the parametric process inside the cavity [15], [16] where A p , A s , A i are the complex field amplitudes of the intracavity pump, signal, and idler waves, respectively. α is the propagation loss and γ is the nonlinear coefficient. z is the coordinate along with the ring. β p,s,i are the propagation constant of pump, signal and idler waves; P is the pump power in the MRR at z = 0 under the undepleted pump approximation.
Since the pump power is much larger than the signal power, only the self-phase modulation (SPM) effect is considered in (1). The asterisk superscript of the complex amplitude represents the complex conjugate. The boundary conditions at the coupling region are given by A s,out = r P in,s + iκA s (L) where P in,p and P in,s are the power of the injected pump and signal lights in the bus waveguide. L = 2πR is the circumference of the ring. The intracavity pump power can be derived by (1) and (4) as is the FE of the pump wave, defined as the intracavity pump field divided by the input pump field [16]. a = e −αL/2 is the round-trip transmission coefficient. ψ p = ϕ p + ϕ N L,p is the total phase of the pump wave, where ϕ p = β p L = n(ω p )Lω p /c is the linear round-trip phase of the pump and ϕ N L,p = γ|A p (0)| 2 L ef f is the nonlinear phase shift induced by the pump wave through the SPM. n(ω p ) is the effective index of the MRR at the pump wavelength and L ef f = (1 − e −αL )/α represents the effective length of the ring considering the presence of losses.
We proceed by writing A s,i = e s,i exp(i(γP + Δβ/2 + β s,i )), and the Δβ = β s + β i − β p accounts for the linear phase mismatch of three involved waves and related to the group velocity dispersion β 2 as Δβ = β 2 (ω s − ω p ) 2 if higher order dispersions are neglected. The signal and idler waves can be written as a 2-vector U (z) = [e s (z), e * i (z)]. By considering the propagation in the MRR as in a waveguide with length equals L and using (2) and (3), the input and output column vector, i.e., U (0) and U (L), respectively, are linked by a transfer matrix describing the parametric processes as [13] where u and v are found as u(L) = [cosh(g) + iq sinh(g)/g] exp(−αL/2), v(L) = [iγP L ef f sinh(g)/g] exp (−αL/2).
Here, g = √ (γP L ef f ) 2 −q 2 is the gain coefficient and q = ΔβL/2 + γP L ef f is the total phase mismatch. The parametric gain G and CE η can be obtained by using (10) and (5) where the ψ s,i = ϕ s,i + ϕ N L,s,i is the total phase of the signal and idler, including the linear contribution ϕ s,i = β s,i L as well as the nonlinear contribution ϕ N L,s,i = 2γP L ef f − q. Equations (11) and (12) are the main results of this paper.

III. DISCUSSION
When assuming that the signal and idler waves coincide with the micro-ring resonances so that ψ s = ψ i = 2mπ, the CE, i.e., (11), can be simplified to: where |FE si | = | κ 1−(ae g )·r · κ 1−(ae −g )·r |. A strong similarity can be found between (13) and (4) in [12], which both show that CE η can be viewed as the product of two parts, the CE in equivalent waveguide η wg and FEs of signal and idler waves. In the low pump power excitation case, η wg_L= γP in,p L · |F E p | 2 and F E s = F E i = iκ 1−a·r ; whereas in the high pump power excitation case, η wg_H = |γP in,p L ef f  Fig. 2(a) and (b), respectively. Fig. 2(a) shows that η wg contributes similarly in the low and high pump cases, which have no explicit feature of the M-shape profile due to the fact that the circumference of the MRR is too short. This is different from the "M" shaped CE curve that appeared commonly in fibers or long integrated waveguides [6], where the propagation length plays an important role. However, Fig. 2(b) shows that |FE si | at high pump power injection is much greater than that at low pump power injection, which is due to the fact that in the high-pump excitation case the parametric gain may compensate the transmission loss so that the effective Q factor of the MRR is enhanced. Moreover, the |FE si | is wavelengthdependent and shows an obvious "M" type CE curve due to the gain coefficient g being related to the frequency interval between the signal and the pump, indicating that |FE si | is the main source of "M" shaped CE curve in the high-Q MRR.
The maximum FE |FE si | at the phase-matched wavelength ω ± = ω p ± −2γP L ef f /β 2 L (i.e., q = 0) can be explicitly written as: Table I summarizes the CEs at waveguides and MRRs in low-power and high-power excitation assuming ψ s,i = 2mπ. In the vicinity of pump wavelength, i.e., Δβ = 0, (11) can be simplified to (4) in [12], leading to a square dependence between CE and |F E p | 2 . On the other hand, the CE in the MRR increases exponentially with |F E p | 2 at phase-matched wavelength, i.e., q ≈ 0, where gain and CE reach maxima. In addition, F E p also affects the |FE si | in an exponential way due to compensation of the intrinsic loss of the MRR, which significantly alters the physical picture of parametric gain and CE compared to the low-pump power case. Therefore, the impact of the field enhancement of the signal and idler wave is far more pronounced in high-pump power situations, especially at phase-matched wavelengths. This is in sharp contrast compared to the waveguide case, where the maximum gain achieved at phase-matched wavelength is mainly contributed from the exponential gain caused by the pump wave. It is also different from the MRR in low-pump excitation case, where the field enhancement of the pump, signal, and idler wave play identical roles.
The gain regime Δω of the OPA is also discussed, as shown in the last column of Table I. Since gain can be achieved between phase-matched wavelengths, the gain regime Δω of the OPA can be roughly estimated by the distance between ω ± , i.e., Δω = |ω + − ω − |.
As shown by Table I, the gain regime in the high-Q MRRs under high-power excitation is broadened by a factor of F E p compared to their waveguide counterpart with the same length L. It also agrees with the well-known conclusion that anomalous dispersion is required to generate the notable parametric gain, i. e., β 2 < 0.
The CE given by (12) and (4) in [12] are shown in Fig. 3(a), where the purple line and yellow dashed line represent our work and [12], respectively. It is clear that the wavelength-independent feature of [12] fails to predict the classical "M" type gain spectrum in the anomalous dispersion regime under a strong pump excitation. The orange dots are obtained from a numerically solved full-map coupled nonlinear Schrödinger equation [14], which tracks the nonlinear process of light in every round trip: Here, A is the complex field amplitude in the ring, including pump, signal, and idler waves. δ is the detuning between the pump and the nearest resonance peak. In the coupling region, the input-output relation follows A out = iκA| z=nL + rA, and n represents the nth round-trip. Standard split-step Fourier method is numerically applied to solve (15), under the situation that detuning δ is zero, which is in good agreement with the CE and parametric gain when the phase terms satisfy ψ s = ψ i = 2mπ, revealing the validity of the derived analytical formulas. Fig. 3(b) shows the gain calculated by (11) as well as full-map simulations.

IV. EXPERIMENT
Finally, to verify the effectiveness of the derived formulas, we experimentally investigate a micro-ring system achieving parametric amplification using a Si 3 N 4 MRR with L = 1.2 mm and the loaded Q is calculated as 4.5 × 10 5 with κ = 0.1243. The experimental results are given in Fig. 4. A CW pump of 21.5 dBm power and a signal of 0.5 dBm power are injected into the bus waveguide. The blue lines and the yellow lines are offresonance and on-resonance. For the situation of on-resonance, the pump wave is finely tuned from a high frequency to approach the resonant frequency. Under the influence of the thermal drift effect of the micro-ring resonators, the pump wavelength is 1550.17 nm and the signal wavelength is 1549,24 nm. The estimated resonant-enhanced intracavity pump power is P = 13.9 W, which is obtained using the methods given in [17]. In the situation of off-resonance while the pump wavelength and signal wavelength are set at 1550.17 nm and 1549,24 nm directly so that without the thermal drift effect. The idler wave is generated at 1551.10 nm when the pump wave is on resonance, with a CE of 3.36 dB and the signal is amplified with a gain of 4.72 dB compared with the situation of off-resonance. The calculated gain and CE are 5.36 dB and 5.24 dB using (11) and (12), slightly larger than the experimental results. The discrepancy may be caused by cascaded FWM.
Due to the alignment challenge induced by thermal shift as well as the limitations given by nanofabrication processes, the validation of our theory is verified only at a given frequency. It is expected that the M-curve of gain and CE spectra can be obtained by more careful device and experimental design.

V. CONCLUSION
In conclusion, we have derived analytical formulas for the parametric gain and CE of the parametric processes in high-Q MRRs, considering actual parameters, including losses, dispersion, nonlinearity, pump power, and coupling coefficients. In particular, the interplay between parametric gain and field enhancement is revealed and discussed. The FE leads to parametric gain first, which in turn compensates for the intrinsic loss of the MRR and leads to further increased FE. The maximum gain achieved at phase-matched wavelength is mainly contributed by the greatly boosted field enhancement of the signal and idler wave. In the end, experiments are carried out to validate the theoretical formulas. It is expected that with the further improvement of integrated photonics, the micro-ring-based OPAs can be leveraged in many scenarios, including all-optical signal processing, optical communications, sensing, etc.