Canonical Four-Wave-Mixing in Photonic Crystal Resonators tuning tolerances and scaling Alexandre Chopin12 Gabriel Marty121 In es Ghorbel1 Gr egory Moille12

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Canonical Four-Wave-Mixing in Photonic Crystal Resonators: tuning, tolerances and

scaling

Alexandre Chopin1,2, Gabriel Marty1,2,∗1, In`es Ghorbel1, Gr´egory Moille1,∗2,

Aude Martin1, Sylvain Combri´e1, Fabrice Raineri2,3, Alfredo De Rossi1

1Thales Research and Technology, Campus Polytechnique,

1 avenue Augustin Fresnel, 91767 Palaiseau, France

2Centre de Nanosciences et de Nanotetchnologies,

CNRS, Universit´e Paris Saclay, Palaiseau, France

3Universit´e Cˆote d’Azur, Institut de Physique de Nice, CNRS-UMR 7010, Sophia Antipolis, France

∗1Now at Saint-Gobain Research Paris, 39 quai Lucien-Lefranc, 93303 Aubervilliers, France

∗2Now at Joint Quantum Institute, NIST/University of Maryland,

College Park, USA and Microsystems and Nanotechnology Division,

National of Standards and Technology, Gaithersburg, USA

∗Corresponding author: alexandre.chopin@universite-paris-saclay.fr

Canonical Four-Wave-Mixing occurs in a resonator with only the required number of modes,

thereby inhibiting competing parametric processes. The properties of the recently introduced pho-

tonic crystal parametric oscillator, Marty et al. Nat. Photonics, 15, 53 (2021), are discussed ex-

tensively. We compare the bichromatic design with other geometries of photonic crystal resonators.

Based on a statistical study over more than 100 resonators and 10 parametric oscillators, robustness

against fabrication tolerances is assessed, performances are evaluated in terms of average values

and their dispersion, and the dependence on the main parameters is shown to follow the theoretical

scaling. The lowest pump power at threshold is ≈40 µW and we show the existence of a minimum

value of the cavity photon lifetime as a condition for parametric oscillation, which is related to three

photon absorption.

I. INTRODUCTION

Non-classical states of light, e.g. entangled photons,

squeezed light are ubiquitous in optical quantum sens-

ing and quantum communication and simulation. These

states are conveniently generated at room temperature

through parametric down conversion in materials with

second order nonlinearity[1]. Resonant enhancement in

optical cavities is used to increase the eﬃciency of these

sources, which have been miniaturized in integrated pho-

tonic circuits. As silicon lacks second order optical non-

linearity, spontaneous Four-Wave-Mixing (FWM) is ex-

ploited as an alternative. Here, two photons from the

pump decay spontaneously into a pair of photons under

the constraint of energy conservation. If the interact-

ing waves are all on resonance with the corresponding

cavity modes, the spontaneous generation rate scales as

R∝(n2

2Q3/V 2)P2with n2the Kerr non-linear index, Q

the quality factor, V the volume of the resonator and P

the pump power [2, 3]. Time-energy entangled photon

pairs have been demonstrated on a silicon chip via FWM

[4] with a microring resonator. By optimizing the non-

linearity of the material and the Q factor, large eﬃciency

can be achieved[5]

Here we discuss a diﬀerent class of resonators, photonic

crystals[6, 7], which diﬀer from ring (and racetrack, disk,

...) resonators in many ways. First, the conﬁnement is

based on Bragg scattering and not total internal reﬂec-

tion. Modes are spatially inhomogeneous and overlap

only in part and, ﬁnally, the modal volume is at least an

order of magnitude smaller than in any other dielectric

resonator. Thus PhC are amenable to a very large nonlin-

ear interaction because they enable a very strong conﬁne-

ment with still potentially large (1M) Q-factors[8].

Nanoscale devices based on Photonic crystal cavities

have been demonstrated: Raman laser[10], electrically

pumped nano-laser integrated on a silicon chip[11], pul-

sating Fano laser[12] and all-optical memory [13]. Their

common point is to operate at very low power (mi-

croWatt regime). The demonstration of optical para-

metric oscillations [14] in a nanoscale PhC cavity with a

threshold of ≈50 µW is particularly interesting in the

context of the scalable generation of squeezed light. In-

tegrated sources of squeezed light[15–17], combined with

a full photonic circuit[18] are used in Gaussian Boson

Sampling[19], a practical conﬁguration to demonstrate

quantum advantage in computing[20]. While the prop-

erties of ring resonators have been extensively studied

and over a variety of photonic platforms, PhC paramet-

ric sources have just been introduced and preliminary yet

promising performances as sources for quantum science

have been reported very recently[21].

In this article we provide a detailed description of the

PhC OPO physics, covering a broad range of operation

and comparing a variety of devices. This work is meant

to provide a comprehensive discussion of this new class of

devices. In section II we will revisit the concept of canon-

ical FWM, meaning FWM in a cavity allowing the inter-

action of only three modes (four in the non-degenerate

case). We will explain why PhC are a suitable choice

and how they diﬀer from ring resonators in this respect,

in particular when considering structural disorder. The

complete model will also be discussed. In section III we

compare the properties of three geometries of PhC multi-

arXiv:2210.04660v1 [physics.optics] 10 Oct 2022

(a)

(b)

Total Internal Reﬂ

ection

Ring/Racetrack Whispering Gallery Photonic Crystal

Wavevector

Frequency

DOS

Defect

localized mode

(c)

(d)

FIG. 1. (a) : Representation of degenerate FWM: energy conservation 2~ωp=~ωs+~ωiand idealized spectrum with the

signal and the idler (ωs,ωi) symmetrically spaced relative to the pump ωp. (b) triply-resonant FWM, interacting photons

(arrows) and cavity modes (black lines). Top : canonical FWM in a resonator with only three spectrally equi-spaced modes;

bottom : multimode resonator with constant FSR, where multiple competing FWM processes simultaneously take place. Partly

inspired from Ref. [9]. (c) conﬁnement in ring, racetrack and whispering gallery resonators is due to total internal reﬂection;

(d) a defect in a dielectric with periodic modulation (Photonic Crystal) induces a localized mode inside the forbidden bandgap

(orange shaded area), a sharp peak in the Density of Optical States (DOS).

mode resonators made of InGaP. In section IV we report

a detailed statistical analysis of a batch of new devices

and show that structural disorder induces uncorrelated

ﬂuctuations of the modes of the same resonator. We also

discuss the tuning mechanism in details. In section V

we compare theory and experiment on 11 OPOs, with

good agreement on threshold and slope eﬃciency. Here,

we elucidate the reason why in InGaP PhC, parametric

oscillation is possible if the Q factor is above a certain

minimum.

II. CANONICAL TRIPLY-RESONANT

FOUR-WAVE-MIXING

Four-Wave-Mixing refers to the exchange of energy

among four optical modes through the ultrafast Kerr

nonlinearity. It is described as the conversion of two

”pump” photons into ”signal” and ”idler” photons, con-

strained by the strict conservation of the energy; e.g. in

the case of a pump-degenerate process 2~ωp=~ωs+~ωi,

thus the signal and idler frequencies are located sym-

metrically relative to the pump, Fig. 1a. The reso-

nant enhancement of FWM requires three (four in the

non-degenerate case) cavity modes with frequencies with

constant free spectral range, such that pump, signal and

idler photons are all on resonance with the cavity. As

an example, this condition is realized in a ring resonator

designed to have a nearly ﬂat dispersion.

Yet, for the purpose of FWM, an ideal resonator would

need to have three or four modes; in contrast, ring res-

onators are over-moded. Let us consider a semiconduc-

tor ring resonator based on a waveguide with eﬀective

index neff = 3.0 and group velocity vg=c/4.0 in

the telecom C-band spectral range (λ≈1550 nm, i.e.

ν= 193 THz). When the free spectral range (FSR)

is set to νn+1 −νn=vg/(2πR) = 400 GHz (hence ra-

dius R= 30µm), the azimuthal order of the modes is

2πνRnef f /c0≈360. This implies that many multiple

FWM interactions are allowed simultaneously, which also

enables microcavity combs[22]. On the other hand, com-

petition between processes is not desirable when the goal

is to maximize the transfer of power from the pump to the

signal and idler in an OPO (Fig. 1b). This is discussed in

recent articles [9, 17] as well as possible strategies. The

case of an ideal triply-resonant FWM has been consid-

ered theoretically and it has been shown that, for some

combination of parameters, operations are stable and the

pump can be entirely converted into signal and idler [23].

Interestingly, it has also been pointed out that this con-

ﬁguration enables the manipulation of the spectral purity

of a laser beam, e.g. a noise eater [24]. Let us refer to

ideal triply-resonant FWM as canonical resonant FWM.

Let us therefore consider a strategy to create a resonator

allowing exactly the required number of high-Q modes

and with controlled frequency spacing. To this aim, let us

consider a class of optical cavities which is radically dif-

ferent from ring, racetrack, whispering gallery resonator,

as conﬁnement is based on Bragg scattering rather than

on total internal reﬂection (Fig. 1c).

FIG. 2. Triplet of modes obtained by combining three single-

mode resonators with coupling strength µand identical an-

gular frequency ω0. They are equally split by √2µin the

tight-binding approximation.

A. Multi-mode PhC resonators

Photonic Crystals are periodic dielectric structures

with complete photonic band gaps, i.e. propagation is

not allowed within some spectral range in any direction

[6]. This condition is satisﬁed if the modulation of the

dielectric permittivity is large enough and along any di-

rection [25]. Deviations from perfect periodicity, e.g. due

to disorder, result into a strong localization of light in

resonant modes [7]. Thus, modes can be created on pur-

pose by introducing defects in the photonic crystal [26],

as shown in Fig. 1d. As a consequence, a resonator with

only the required modes is feasible by introducing the ap-

propriate number of defects. Yet, there are considerable

diﬃculties to be considered.

First, periodic structures with a complete band gap are

very diﬃcult to fabricate. However, it has been demon-

strated that a periodic modulation in a thin slab of

high index dielectric, e.g. Silicon, can be engineered

to host high-Q resonances (Q≈107) by carefully min-

imizing out-of-plane radiation[8, 27]. The second chal-

lenge is the control of the FSR in multi-mode PhC res-

onators. Ultra-eﬃcient Raman lasing was demonstrated

by matching the spacing of two modes in a high-Q PhC

resonator to the Raman peak in crystalline Silicon[10].

The control of more modes is increasingly challenging.

Coupled PhC cavities have been considered to create

multi-mode resonators aiming at dispersion control and

slow-light [28, 29], four-wave mixing and correlated pho-

ton pairs[30, 31], demonstrating the optical equivalent of

Electromagnetically Induced Transparency[32] and time-

parity symmetry breaking[33].

According to the Tight-Binding (TB) approximation,

three identical resonators coupled in a chain with

strength µcreate a triplet with spacing ν±1−ν0=

±√2µ(Fig. 2), which exactly corresponds to what is

needed for canonical FWM. This has been experimen-

tally demonstrated by combining three nanobeam PhC

Detuning

FIG. 3. Calculated normalized stimulated FWM as a func-

tion of the normalized probe detuning, depending on the dis-

persion ∆2ν.

resonators[34]. Yet, the TB approximation is not ade-

quate here: mode splitting is not symmetric because of

the dispersive nature of the coupling in PhC [35]. Be-

sides, this is also true in the case of ring resonators [36].

Thus, even in the case of a perfect fabrication of the

intended geometry, there will be a non-zero dispersion

∆2ν= (ν1−ν0)−(ν0−ν−1). Triply resonant FWM is

still possible if the dispersion is small compared to the

spectral width of the resonance Γ/2π, with Γ the cav-

ity photon decay rate. Fig. 3 shows that the stimulated

FWM conversion eﬃciency, computed using eq.10, de-

creases by one order of magnitude when |∆2ν| ≈ Γ/2π.

Thus, the consequence of misalignment is to force the

choice of a lower Q factor for the resonator according to

Fig. 3. In Ref. [34], the Q factor is about 4000, which is

low enough to ensure tolerance with respect to the dis-

persion and fabrication disorder.

B. Implications of the structural disorder in PhC

resonators

Fabrication imperfections, e.g. surface roughness, in-

homogeneities in the material, in the lithography or in the

etching process, induce ﬂuctuations in the resonances.

This has been investigated in high-Q Silicon PhC res-

onators. Here, the standard deviation is estimated to

about 40 GHz, which is considered to be a lower limit.

In fact, Silicon PhC technology has demonstrated the

record high Q-factor for PhC resonator and, therefore, of-

fers state-of-the-art fabrication imperfections [8]. There,

the standard deviation describes ﬂuctuation of a single

mode in diﬀerent resonators.

Since modes have inhomogeneous and partially overlap-

ping spatial distribution in a multi-mode PhC resonator,

it is expected that the ﬂuctuations of their frequencies

are mutually uncorrelated. Therefore the FSR and the

dispersion ∆2νwill essentially have the same standard

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CanonicalFour-Wave-MixinginPhotonicCrystalResonators:tuning,tolerancesandscalingAlexandreChopin1;2,GabrielMarty1;2;1,InesGhorbel1,GregoryMoille1;2,AudeMartin1,SylvainCombrie1,FabriceRaineri2;3,AlfredoDeRossi11ThalesResearchandTechnology,CampusPolytechnique,1avenueAugustinFresnel,91767Palaiseau,...

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Canonical Four-Wave-Mixing in Photonic Crystal Resonators tuning tolerances and scaling Alexandre Chopin12 Gabriel Marty121 In es Ghorbel1 Gr egory Moille12

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