Photonic Crystal Optical Parametric Oscillator

Abstract

We report a new class of Optical Parametric Oscillators, based on a 20μm-long semiconductor Photonic Crystal Cavity and operating at Telecom wavelengths. Because the confinement results from Bragg scattering, the optical cavity contains a few modes, approximately equispaced in frequency. Parametric oscillation is reached when these high Q modes are thermally tuned into a triply resonant configuration, whereas any other parametric interaction is strongly suppressed. The lowest pump power threshold is estimated to 50 - 70μW. This source behaves as an ideal degenerate Optical Parametric Oscillator addressing the needs in the field of quantum optical circuits, paving the way to the dense integration of highly efficient nonlinear sources of squeezed light or entangled photons pairs.

Miniaturization of devices has been a primary objective in microelectronics and photonics for decades, aiming at denser integration, enhanced functionalities and drastic reduction of power consumption. Headway in nanophotonics is currently linked to the progress in concepts and technologies necessary for applications in information and communication[1], brain-inspired computing [2], medicine and sensing [3] and quantum information [4]. Amongst all nanostructures, semiconductor photonic crystals (PhCs) [5] occupy a prominent position as they enable the fabrication of quasi ultimate optical cavities [6]. Low threshold laser diodes[7, 8] or Raman lasers[9], low power consuming optical memories[10], efficient single photon sources [11] or single photon quantum gates [12] are impressive examples of their capabilities. The PhC implementation of an important class optical sources, namely Optical Parametric Oscillators (OPO), is missing.

An OPO emits coherent light relying on the ultrafast nonlinear response of matter for the stimulated emission of photon pairs. OPOs can generate light within a spectral range only limited by the transparency of the nonlinear material. As the generated photons are correlated, OPOs are also sources of nonclassical light for quantum optics[13, 14] and quantum computing[15], OPOs were originally available only as solid state optical devices, tending to be bulky and expensive. About 15 years ago, the progress in the fabrication of high-Q resonators led to microcavity OPOs[16, 17] and the ensuing birth of a new domain in nonlinear photonics. They have been constantly improved under virtually all respects and have been recently integrated with laser diodes into very compact packages[18, 19].

Microresonators are based on disks or rings, where the confinement results from the total internal reflection. Essential here is engineering the dispersion of the propagating waves, in order to control the nonlinear interaction of a large number of modes. This enables soliton combs [20] or the ultra broadband generation of light[21–23], to make a few examples.

On the other hand, PhC cavities are fundamentally different devices that confine a small number of modes within a photonic bandgap[24, 25]. Moreover, higher resonances with diffraction-limited mode volumes ≈ (λ/n)³, with n the refractive index, are possible in a PhC cavity[5, 6], which would lead to a favourable figure of merit Q ²/V for lowering the power threshold for parametric interactions [26], The possibility of a PhC OPO was considered theoretically more than a decade ago[27, 28] and experimental attempts to achieve efficient parametric interaction have been made [29, 30], yet demonstrating OPO revealed extremely challenging.

Photonic Crystal OPO Technology

Our PhC OPO harness triply resonant degenerate Four Wave Mixing (FWM) (Fig. 1a). For this Kerr optical nonlinear process, energy conservation dictates the relation between the pump and the generated photons $(2{\omega }_p={\omega }_i+{\omega }_s),$ meaning that the new angular frequencies ω_i and ω_s are generated symmetrically with respect to the pump ω_p. Thus, resonant enhancement is only possible if three cavity eigenfrequencies (ω ₀, ω ₊, ω ₋) are rigorously equispaced. This condition is in general not satisfied in PhC resonators. Triply resonant parametric interaction was observed in a system made of three coupled PhC nanocavities, however with a limited efficiency due to low Q factor[29] or not well controlled frequency spacing[30]. In fact, the control of the frequency spacing is ultimately limited by structural disorder: even in state-of-the-art PhC technology, resonances deviate from their design value by about 40 GHz [31], For this reason, in absence of a tuning mechanism, the eigenmode linewidth must be large enough to allow triply resonant FWM. This corresponds to Q < 5 × 10³, clearly hindering the opportunity offered by PhC [6].

Fig. 1. Triply resonant PhC cavity.

a Concept of the triply-resonant degenerate OPO where resonant modes respect the energy conservation $2\hslash {\omega }_0+\hslash {\omega }_{+}+\hslash {\omega }_{-};$ b calculated Hermite-Gauss modes corresponding to an effective parabolic potential for photons (gray solid lines) and eigenmode frequencies and field amplitudes (color lines) of the bichromatic photonic crystal cavity; the filled curves denote a triplet of interacting modes; c Device layout with periods a and a′ and access waveguide and SEM image of the InGaP membrane; d measured frequencies of the first 4 modes (relative to cold resonance ${\overline{\omega }}_0$) as a function of the pump offset Δ_o and linear fit (dashed); e corresponding eigenfrequency intervals (FSR).

In this work, we demonstrate a PhC OPO operated at ultra-low power. This result was achieved owing to three key features: first, we designed the cavity to have equispaced eigenfrequencies, then, we introduced a differential thermo-refractive tuning mechanism to compensate for the residual spectral misalignment caused by fabrication imperfections and, finally,we used In_0.5Ga_0.5P III- V semiconductor as the constitutive material of our cavity. The large electronic bandgap of this material enables the mitigation of Two Photon Absorption (TPA)[32] which otherwise clamps nonlinear conversion efficiency below parametric oscillations.

The PhC cavity[33] is designed to create an effective parabolic potential for the optical field, Fig. 1b, such that eigenfrequencies are equispaced and the eigenmodes correspond to Gauss-Hermite (GH) functions. The design (Fig. 1c) consists of a suspended membrane with a regular hexagonal lattice of holes with period a, except for a line of missing holes, where light can propagate. There, we introduce a bichromatic lattice [34], by modifying the period a′ of the innermost row of holes. The localization of the field and the Free Spectral Range (FSR) between higher order modes are controlled by the commensurability parameter a′/a. Choosing a′/a = 0.98 corresponds to a FSR about 400 GHz and the field envelopes are very close to GH functions, Fig. 1b. The optical access to the modes is provided by a single-ended waveguide on the side terminated with a mode adapter to maximize coupling to an optical fiber [35].

Operation

Our fabrication process ineluctably induces fluctuations on the targeted eigenfrequencies that result in a misalignment (50 GHz on average) far greater than the resonance linewidth, below 1 GHz (details in Supplementary Section II.C). We use thermal tuning to compensate for this deviation. Thermal control of the parametric interaction has been implemented by individually controlling two coupled resonators[36] or exploiting the dispersive nature of the thermo-refractive effect [23]. In our approach, a localized dissipation induces a spatial gradient in the refractive index of the material. In general, the spatial overlap of each mode with the perturbation of the refractive index will be different, and so will be the corresponding spectral shift of the resonances. Local heating can be realized through the projection on the sample surface of a patterned incoherent pumping beam[37], this method being limited by the precision of the projection system. In our work, this effect is automatically achieved when the pump mode “0” is by resonantly excited, as the energy dissipated induces a temperature gradient. Importantly, Fig. 1b shows that the GH modes differ markedly in their spatial energy distribution, such that the effect of the temperature gradient on the differential spectral shift of the modes is enhanced. Details are given in Supplementary Section II.A.

In order to demonstrate thermal tuning, we have performed a pump-probe measurement of the reflection spectra as the pump laser at ω_p and constant power 0.8 mW is swept from blue to red across the resonance. This is achieved using an optical homodyne technique (methods) using an additional swept laser to detect with high accuracy the pump and the ω ₊ and ω ₋ modes involved in the FWM process. As shown in Fig.1d, the pumped mode “0” redshifts from ${\overline{\omega }}_0$ (cold) to ω ₀ (hot) while the other modes shift differently as expected. The misalignment of the hot cavity (corresponding to the dispersion) $2{\Delta }_x=2{\omega }_0-{\omega }_{-}-{\omega }_{+}$ is deduced from these measurements and is found to follow $2{\Delta }_{\chi }=2{\overline{\Delta }}_{\chi }+0.48{\Delta }_0$ to the first order, where ${\overline{\Delta }}_{\chi }$ is the misalignment of the “cold” cavity and ${\Delta }_0={\omega }_p-{\overline{\omega }}_0$ is the pump offset. This shows that an originally mismatched triplet can eventually be aligned by adjusting the pump frequency, which is clearly observed in Fig.1e where the two considered FSRs are equalized (crossing of blue and red lines).

We now investigate how this tuning mechanism affects the parametric gain in stimulated FWM measurements, where an additional laser (“signal”) at ω_s is swept across the resonance at ω ₋ while the pump frequency is fixed and an idler wave is generated at ${\omega }_i=2{\omega }_p-{\omega }_s.$ As a matter of fact, the stimulated efficiency ${\eta }_{\chi }=P_i/P_s,$ defined as the ratio between the output idler power (P_i) over the input signal power (P_s), is directly related to the parametric gain and the loss of the cavity. In fact, parametric oscillation is reached when the gain is large enough to compensate for the loss, and the the stimulated efficiency diverges (Supplementary Section Eq. 8). The measured values of η_χ (details in Supplementary Section II.D) for a sample with an average Q-factor $Q_{avg}={(Q_0^2{Q}_{-}{Q}_{+})}^{1/4}$ about 7 × 10⁴ are displayed on Fig. 2a as a function of the pump offset for an input pump power of 700 μW. The efficiency η_χ peaks at 0.4 % (-25dB) when ∆₀/2π = -110GHz, in agreement with the triply resonant configuration predicted by our thermal tuning measurements shown Fig. 2b. To support these results, an analytical model (see Supplementary Section I) is derived and η_χ, in the limit of undepleted pump and low parametric gain, becomes:

$${\eta }_{\chi }={\eta }_{\chi }^{\left(max\right)}ℒ{\left(\frac{{\delta }_0}{{\Gamma }_0}\right)}^2ℒ\left(\frac{{\delta }_{-}}{{\Gamma }_{-}}\right)ℒ\left(\frac{{\delta }_{+}}{{\Gamma }_{+}}\right)$$ (1)

Where ${\delta }_0={\omega }_p-{\omega }_0,{\delta }_{-}={\omega }_s-{\omega }_{-,}{\delta }_{+}={\omega }_i-{\omega }_{+}$ are the frequency detunings of the pump,signal and idler with respect to the hot resonances and Γ₀ and Γ₋, Γ₊ are the photon damping rates in the corresponding modes $(\Gamma =\omega /Q).$ ℒ(x) is the Lorentzian function. The maximum achievable efficiency ${\eta }_{\chi }^{\left(\mathit{\text{max}}\right)}$ is obtained when the triplet is aligned and when the generated frequencies match the resonant modes $({\delta }_0={\delta }_{-}={\delta }_{+}=0).$ The calculated map of η_χ versus the pump offset and probe detuning δ ₋ reveals two local peaks merging into an absolute maximum of efficiency, corresponding to a perfectly aligned cavity (Fig. 2b). Remarkably, the measurements are in quantitative agreement with our model, as per inset of (Fig. 2a, when the parameters are either calculated or measured, and the intracavity power adjusted by less than 10% to account for experimental uncertainties (more details in the Supplementary Section III.A). The thermal tuning technique allows reaching a larger FWM efficiency as resonators with larger Q-factors can still be tuned to form an equispaced triplet. We performed stimulated FWM on a sample with a larger average Q = 1.5 × 10⁵. It results in a drastic improvement of the efficiency that rises up to 26 % (−5.8dB) with an on chip pump power of 80 μW, Fig. 2c and Fig. 2d. We also note that the dependence of the efficiency on the probe detuning, Fig. 2c, is very sharp, because of the triple resonant enhancement of FWM.

Fig. 2. Stimulated Four Wave Mixing.

a Measured stimulated FWM efficiency η_χ as a function of the pump offset Δ₀ and probe detuning δ ₋; (Inset) Comparison with the theory (black line) for the red curve, (see Supplementary Fig. S3 for full comparison) b Calculated false color map of the efficiency η_χ of stimulated FWM as a function of the probe detuning δ ₋ and pump offset with corresponding effective mode temperature rise ΔT ₀ for mode 0. The white lines represent the poles of eq. 1. c Stimulated FWM in a resonator with larger Q_avg. Raw spectra centered on the pump ω ₀ as a function of the probe detuning δ ₋ spectra for tuned (red) and detuned (blue) probe; d reflected probe (blue) and idler (red) power vs. probe detuning.

Optical Parametric Oscillation

In these samples, parametric oscillation is not observed, meaning that the maximum realizable parametric gain is not sufficient to compensate for cavity loss. This value is entirely determined by the cavity optical properties (Q, mode volume, nonlinear cross-section, material nonlinearity) and the pump power level required to align the three modes. Its value is therefore unique for each triplet of modes. Eventually, parametric oscillation is demonstrated for a resonator with $Q_{avg}\approx 2.5\times {10}^5.$ The sample is pumped with an on-chip power level below 200 μW, which is enough to align a triplet of adjacent resonances. As the pump offset ${\Delta }_0/2\pi <-170\text{GHz},$ the red ω ₋ and blue ω ₊ signals emerge from noise $(-73\text{dBm}=50\text{pW)},$ as shown in Fig.3a. When approaching -175 GHz, they abruptly increase by four orders of magnitude, a clear indication of an oscillation threshold. Interestingly, the pump offset, is, to a good approximation directly proportional to the energy stored in the pump mode as the spectral shift is induced by linear absorption (see Supplementary Section II.A). Therefore, the result can be cast in a more familiar representation by estimating the equivalent pump power circulating in the cavity P _c,0 in Fig. 3b. Above threshold $P_{c,0}>P_{th}=175\mu W,$ the on-chip generated power increases linearly with ${\eta }_{\pm }(P_{c,0}-P_{th})/2,$ with η _± the escape efficiency[38] from the side modes (defined as ${\kappa }_{\pm }/{\Gamma }_{\pm },$ see also Supplementary Tables I and III). More than 50% of the excess pump power is converted. Strikingly, the threshold expected from our model is 170 μW (see Supplementary Table III). Experiments have been repeated with three other candidates, labelled a5, a7 and a12, having suitable parameters: $Q_{avg}>2\times {10}^5$ and $0<{\overline{\Delta }}_{\chi }/2\pi <50GHz.$ As shown in (Fig. 3c-e, they all demonstrate OPO over a larger operating range, generate more power and have lower threshold. In particular, the estimated is P_th = 50 – 70μW for sample a5. The methodology, the reproducibility of the results and the estimate of the uncertainties is broadly discussed in the Supplementary Section III.B and it is shown that the results are consistent with the prediction based on the simple equation:

$$P_{th}=\frac{{\epsilon }_r{V}_{\chi }\omega }{8c_0{n}_2}\frac{1}{Q_{avg}^2}\frac{1}{{\eta }_0}$$ (2)

the measured Q_avg arid the cavity coupling efficiency of the pumped mode ${\eta }_0={\kappa }_0/{\Gamma }_0.$ We also note that the corresponding energy stored in the cavity is below 30 fJ, and even about 10 fJ in the best device (Supplementary Section III.B). Energy at threshold is set by the loss rate at the side modes, hence to Q ₊ Q ₋. Importantly, if the energy in the cavity is below the level at which nonlinear absorption takes over linear absorption, then a possibly detrimental thermal runaway is avoided, which explains why OPO was observed only at large Q (see Supplementary section II.A and Table III).

Fig. 3. Parametric Oscillation.

a Raw optical spectrum (cavity b8, resolution 4 GHz, centered at the pumpfrequency) as the pump offset is changed. The threshold is overcome as the intra-cavity pump energy increases. Markers represent the raw power on the red and blue side, solid black line is thespectrum at maximum OPO emission; b On-chip power in the blue ω ₊ and red ω ₋ (makers) as a function of the pump offset and equivalent pump power in the cavity P _c,0 with a threshold of 175 μW for cavity b8, lines indicate the slope efficiency c 50 μW (cavity a5), d 100 μW (cavity a12), e 180 μW cavity (a7); f OPO pump threshold as a function of the footprint compared with state the art in microring and racetrack resonators made of different materials (details and references are in the Supplementary Table IV).

Discussion

In order to set this result in the context of integrated OPO, we consider the power at threshold P_th as a function of the device footprint (Fig. 3f). The performance of our ultracompact system based on interacting standing waves is already comparable to that of recently demonstrated microring and racetrack OPOs which exhibits power thresholds between 30μ,W and 25 mW. Let us note that the resonator footprint here is defined as the smallest area that ensures the confinement of the modes. It corresponds in our case to 20x6 μm², including the coupling waveguide shown in Fig.1c.It is apparent that low power threshold of our PhC OPO results from the strong confinement of the interacting modes, the large nonlinearity of semiconductors, compensating the moderately large Q factor, as compared to crystalline, Silicon Nitride or Silica resonators. Confinement is an essential point here. Taking into account the spatial overlap between the interaction modes, which corresponds to the phase matching condition in waveguides, leads to the nonlinear interaction volume (eq. 2), which is here V _η = 5.7μm³. This is about factor 10 smaller than the corresponding interaction volume deduced for the ring resonator V_η,ring = 1.54 × 2πA_effL = 50μm³ (Supplementary section III.C). The significance of this result is having made a step towards an idealized device with the smallest possible interaction volume. So far, lower power thresholds were observed very recently [39] in Al GaAs based microring (≈ 30μW), owing to the larger the nonlinearity of the material and much higher Q-factors. Yet, lower values (down to 5 μW) are only reported in crystalline resonators with ultra-high Q factor and based on the second order nonlinearity[40]. Considering that current state-of-the art PhC cavities[6] exhibit Q > 10⁷, OPOs with power thresholds approaching the μW level can be realistically considered if the Q factor of our resonators could be improved to be larger that 10⁶. This could be achieved using surface passivation techniques such as in [39] which we have not yet used.

As in any other laser source, power threshold is not the sole figure of merit. Also the overall energy efficiency transfer from the pump to the side modes is crucial. In our case, the estimated total (signal + idler) on-chip generated power is about 20 μW, as shown in Fig. 3(d,e), leading to a conversion efficiency which is about 15% of the coupled pump power. This value can be compared to the one of 17 % very recently reported in AIN ring resonators[41], which are operated at much larger power (10 mW) and exploit the η ⁽²⁾ nonlinearity. Moreover, the measured slope efficiency (signal+idler) is about 50% in almost all our measurements, only limited by the escape efficiency η± (Supplementary Table III). In this respect, an increase of the Q factor would offer more room for increasing η± without sacrificing the power threshold. Finally, it is interesting to point out that our OPO behaves as a pure degenerate parametric system with only three cavity modes interacting. Simultaneous alignment of more than one triplet is very unlikely. This is apparent in Fig. 1e, where another triplet (x, + and 0) is aligned at a different pump offset Δ₀ ≈ 50 GHz. While this still allows non-degenerate stimulated FWM, this process highly inefficient and might be neglected in most cases. Moreover, we show that other modes do not interact at all (Supplementary section III.B). For the same reason, resonant nonlinear contributions such as Raman can be ignored, unless the resonator is deliberately designed for, as in Ref. [9].

It has been recently pointed out that a pure parametric system is highly desirable for some applications. In particular, achieving squeezing on chip using a degenerate OPO has drawn a lot of interest recently[38, 42–44], Moreover, it has been shown very recently that a network of coupled degenerate OPO could be implemented on chip to implement a coherent Ising machine [45], A different idea has also been proposed recently to use a degenerate OPO, as a "noise eater" 46], In ring resonator, there are strategies to control higher order interactions, for instance by inducing sidewall corrugation[47] or adding an auxiliary resonator [36, 38] or implementing narrowband FWM by implementing a special dispersion profile[21, 23]. Here, the devices operates naturally as a degenerate triply resonant OPO, although it could in principle be possible to achieve efficient cascaded interactions with some additional engineering[48] and tuning. In conclusion, we demonstrated, to the best of our knowledge, the parametric oscillation regime in a Photonic Crystal Cavity. In addition to the low operating pump power (50-70μ W), the efficient conversion to the sidebands (combined slope efficiency ≈ 50%) and the absence of competing parametric interactions, this source is amenable to the incorporation within sophisticated optical circuits [49]. This demonstration opens up exciting avenues for building an integrated all semiconductor plat form to optically generate and process both classical and quantum data.

Methods

Fabrication

The cavities are suspended membranes made of Indium Gallium Phosphide lattice-matched to GaAs. The photonic crystal is created using e-beam lithography, dry etch of the hard mask, inductively coupled plasma etching of the holes and wet etching of the underlying material substrate to release the membrane[50]. The large electronic gap (1.89 eV) prevents two-photon absorption when operating the telecom spectral band, while the residual absorption rate ${\Gamma }_{abs}=2\pi \times 10MHz$ is very low[51].

Optical Measurements

Optical measurements have been performed on a temperature stabilized sample holder position stage and the two-way optical access is provided by a microscope objective lensed fiber, actuated by a 3-axis nanopositioner stage, and a circulator. A tunable laser source is connected to the sample through a variable attenuator and the output is directly fed to the Optical Spectrum Analyser.

High-resolution optical measurements are performed using an optical homodyne technique. A continuously swept laser is used to scan an Michelson interferometer where the sample replaces one of the mirrors. The detected fringes carry information about the electric field, an therefore Fourier analysis can be used to extract both the phase and the amplitude of the reflectivity, perform advanced noise filtering and also deduce the impulse response of the device. Further details are in Suppl. Inf. in Ref. [33]. This system has some features which are very convenient for our study. First, the spectral resolution is about 20 MHz although the scan is fast (less than 10s over a spectral range spanning 8 THz), which is adequate to high-Q resonator. Second, the sensitivity is very high, thus we can use a very weak prove (about 100 nW) to ensure a purely linear response and, finally, this allows CW pump-probe analysis, as the pump is not detected as it is detuned from the local oscillator.

Extended Data

Extended Data Fig. 1. Triply-resonant Cavity.

a: Representation of a resonator coupled to a single-ended waveguide in the time-dependent coupled mode theory with definition of field in the waveguide s and in the cavity a, internal Γ abs and radiation loss Γ rad and waveguide coupling κ; b angular frequencies ω for the normal mode and excitation waves, “cold” resonances ω and detuning δ. c Linear scattering spectrum measured using OCT revealing the ω ± and ω0 modes d and corresponding calculated modes.

Extended Data Fig. 2. Thermo-Optic Tuning.

Description of the tuning measurement, with a pump pulling mode ω0 and detecting all the resonances a; calculated profile of the temperature (solid filled) and the dissipated energy (dashed) b and corresponding 2D map c; measured spectral shift as a function of the energy stored in the mode (markers) and fit; d estimated absorption rate including nonlinear absorption e.

Extended Data Fig. 3. Reflectivity Measurement and Model.

Linear spectral characterization of a resonator using OCT. a Extracted temporal trace revealing a narrow peak (refection from the end facet) and a broad dispersive peak (reflection from the cavity); b corresponding spectrum superimposed with the measurement of the reflection using a direct (non heterodyne) detection; c model representing reflection at the PhC coupler (=waveguide end facet) and from the cavity; d fitted reflection from cavity 5 and e cavity 7 from which the coupling factor K is extracted.

Extended Data Fig. 4. Measured Stimulated FWM efficiency vs. Theory.

Measurement of the stimulated FWM efficiency ηχ as a function of the pump offset ∆₀ and probe detuning δ₋; the corresponding measured FSR is in the inset a; comparison with the model (inset with colored frame); b max(ηχ) as a function of the pump offset, experiment (symbols) and theory (solid line).

Extended Data Fig. 5. Calibration of the OPO measurement.

OPO measurement on cavity a5. a on-chip power in the sidebands as a function of the effective pump power P_c,0 (note that the idler ωX is rescaled); b corresponding reflected pump power and calculated reflection when the pump is off-resonance (solid line), on chip pump power is 72μW. Same horizontal axis.

Extended Data Fig. 6. OPO threshold.

OPO pump threshold as a function of the Q factor compared with the state of the art in microring and racetrack resonators made of different materials. Details and references are in the Supplementary Table IV.

Data Avaliability

The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.