Fast dispersion tailoring of multi-mode photonic crystal resonators Francesco Rinaldo Talenti12 Stefan Wabnitz13 In es Ghorbel2

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Fast dispersion tailoring of multi-mode photonic crystal

resonators

Francesco Rinaldo Talenti1,2,∗, Stefan Wabnitz1,3, In`es Ghorbel2,

Sylvain Combri´e2, Luca Aimone-Giggio2, Alfredo De Rossi2

1Dipartimento di Ingegneria dell’Informazione, Elettronica e Telecomunicazioni,

Sapienza University of Rome, 00184 Rome, Italy.

2Thales Research and Technology, Campus Polytechnique,

1 Avenue Augustin Fresnel, 91767 Palaiseau, France.

3CNR-INO, Istituto Nazionale di Ottica,

Via Campi Flegrei 34, 80078 Pozzuoli (NA),

Italy. ∗Corresponding author: francescorinaldo.talenti@uniroma1.it

arXiv:2210.12017v1 [physics.optics] 21 Oct 2022

Abstract

We introduce a numerical procedure which permits to drastically accelerate the design of mul-

timode photonic crystal resonators. Speciﬁcally, we demonstrate that the optical response of an

important class of such nanoscale structures is reproduced accurately by a simple, one-dimensional

model, within the entire spectral range of interest. This model can describe a variety of tapered

photonic crystal structures. Orders of magnitude faster to solve, our approach can be used to

optimize certain properties of the nanoscale cavity. Here we consider the case of a nanobeam

cavity, where the conﬁnement results from the modulation of its width. The proﬁle of the width

is optimized, in order to ﬂatten the resonator dispersion proﬁle (so that all modes are equally

spaced in frequency). This result is particularly relevant for miniaturizing parametric generators

of non-classical light, optical nano-combs and mode-locked laser sources. Our method can be easily

extended to complex geometries, described by multiple parameters.

I. INTRODUCTION

The nonlinear interaction among several resonant ﬁelds in an optical resonator leads to

eﬃcient Raman and Brillouin scattering, three and four-wave-mixing, optical parametric

oscillation[1], laser mode locking and frequency comb generation[2]. Scaling down the size

of optical resonators implies that the optical power level for triggering nonlinear eﬀects

decreases as V−1or V−2, where Vis an eﬀective volume of the spatial distribution of the

interacting ﬁelds. In the context of photonic integration, the decrease of the power budget

is of paramount importance.

Nanoscale optical resonators such as photonic crystals are able to conﬁne light within V≈λ3,

i.e., a wavelength-sized volume, with a photon decay time, or interaction time, well above

1 ns (i.e., the cavity quality factor Q106). Owing to these properties, it has been

possible to demonstrate nanoscale lasers[3–6], Raman sources[7] and, more recently, optical

parametric oscillators[8], all operating with a power supply (optical or electrical) in the µW

range. Yet, a major challenge remains in achieving the nonlinear interaction of multiple

longitudinal modes, as it occurs in mode-locked lasers or in micro-combs. While ring or

microdisk resonators naturally provide the necessary, nearly frequency equispaced set of

cavity resonances, achieving the same condition in nanoscale resonators is notoriously a

nontrivial task. On the other hand, nanoscale resonators could, in principle, be designed in

a way that a speciﬁed number of modes, starting from the fundamental, and only these, are

allowed to take part to a nonlinear interaction. This unique property implies not only that a

much higher degree of control on power transfer among modes (which is crucial in quantum

and signal processing applications[9]) can be achieved, but also leads to maximizing the

interaction eﬃciency. This is because, in the typical conﬁguration of a nanoscale resonator,

the lowest order modes are also the most tightly conﬁned. Moreover, in a mode-locked

nanolaser, the control of the interacting modes would enable a favorable scaling of repetition

rate vs. the size of the device[10].

It has been shown that some speciﬁc designs of a photonic crystal cavity lead, for some

set of parameters, to frequency equispaced eigenmodes; moreover, their mode envelopes are

described by Hermite-Gauss functions. This suggests that, within a certain spectral range,

the complex photonic crystal structure can be well approximated by a quantum-mechanical

harmonic oscillator model[11, 12]. It has also been shown that post-fabrication trimming

is eﬀective in correcting for fabrication tolerances, thereby demonstrating an almost perfect

alignment of the cavity resonances[13]. Yet, a systematic design approach for generating a

given number of equispaced modes, or, more generally, with a prescribed dispersion proﬁle,

while at the same time maximizing the radiation-limited Q-factor, is still missing, while

brute-force methods are extremely ineﬃcient.

Finding a cavity geometry, or more generally, a physical system whose response to an input

excitation corresponds to a well-deﬁned target function, e.g., a spatial distribution of the

dielectric permittivity such that the electromagnetic ﬁeld has prescribed resonances, belongs

to the class of inverse problems, which are notoriously diﬃcult to solve. Yet, the progress

of nanofabrication techniques has motivated the development of powerful methods such as

topological optimization (TO) [14] and inverse design (ID) [15]. The common feature of these

two approaches is that their result is a spatial distribution of ε(x), rather than an optimized

set of parameters for a pre-deﬁned geometry. These methods are therefore able to create

novel geometries, hence the reference to design. Moreover, automatic diﬀerentiation[16] and

the adjoint method[17] enable a very eﬃcient computation of the gradient, which is required

in the iterative search of the optimum distribution, even in the presence of nonlinearity.

Here we follow a radically diﬀerent approach, which is arguably more suited for the class of

problems under consideration. This is motivated by the fact that the geometries of nanoscale

(a)

(b)

(c)

(d)

FIG. 1. Common design of high-Q res-

onators based on gentle conﬁnement: (a) ta-

pered Distributed Feedback Grating[18], (b) 1-

D nanobeam with parabolic width[19], (c) 1-D

nanobeam bichromatic[20], and (d) 2-D bichro-

matic resonator[11], and corresponding taper-

ing parameter ∆.

resonators with the largest experimentally reported Q factors[18, 21–24] are still based on

the principle of gentle conﬁnement[25]. In other words, these nanostructures are essentially

periodic, with an adiabatic tapering of some parameters, i.e., a gentle change of the radius

of the holes, the period or the magnitude of a ”dislocation” defect, etc.. We note that more

aggressive design strategies, including TO or ID, have instead been considered for diﬀerent

tasks, e.g., for maximizing light-matter interactions in single-mode resonators[16, 26].

Let us restrict our search to a family of structures which can be described by means of

a periodic pattern ε(x,∆) that depends on a control parameter ∆, which is supposed to

adiabatically vary in space (i.e., gently). Some examples of such geometries are given in

Fig. 1. The crucial point is that it is possible to map the three-dimensional (3D) Maxwell

equations (ME) into an equivalent system of one-dimensional (1D) equations, which will be

referred to as the reduced model (RM). Remarkably, the relative precision of the resonances

predicted by the RM turns out to be at least as good as the precision of the direct numerical

solution of the 3D ME. The search of the desired optimal spatial dependence of ∆ will be

performed by using any suitable optimization method, leveraging on the extremely faster

solution of the RM, when compared the direct solution of the 3D ME. The RM in itself

only requires a single direct solution of the 3D ME, for building an initial approximation

of the structure. Subsequent applications of the RM are used, in order to reﬁne the ﬁrst

approximation. As we shall see, in total only three 3D solves are suﬃcient for obtaining

a design that matches our target, with an accuracy that is equivalent to that of directly

solving the 3D ME, but with a comparatively much larger number of iterations.

Hereafter we will ﬁrst discuss the derivation of the RM, then we will formulate a design target

followed by the introduction of the optimization procedure, including model calibration.

Finally, we will discuss possible applications and generalizations of the model.

II. REDUCED MODEL FOR A PERIODIC PHOTONIC CRYSTAL

(a)

0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4

180

200

220

240

260

/2 (THz)

g,2

vg,2

0/2

0.4 0.2 0.0 0.2 0.4

k k0(units of /a)

100

200

300

(b)

/2 (GHz)

FIG. 2. (a) Dispersion diagram of a periodic

structure (see inset) with w= 450nm and pe-

riod a= 465nm, centered in the Kpoint of

the reduced Brillouin zone (k0=π/a). The

reduced model considers coupled forward and

backward waves (dashed lines), generating the

valence and conduction bands (solid lines). The

ﬁlled circles represent the valence band calcu-

lated by periodic 3D ME. (b) The correspond-

ing residuals σ/2πof the ﬁt.

The reduced model is inspired by the so-called k·pmethod [27], which is used in solid

state physics to model the electronic band structure of crystals. The main idea of the

method is to describe the dispersion relation of the electronic bands (i.e., electron energy vs.

wavevector k) through a suitable algebraic equation, which is built upon the eigenfunctions

of the exact Hamiltonian at the bands extrema (at points of high symmetry, e.g., k= 0, or

gamma point). Within a range of energies of interest, the dispersion relation is extrapolated

from the gamma point by treating the k·pterm as a perturbation. In this way, the com-

plexity of solving the full Schr¨odinger equation for the crystal is reduced by using a much

simpler model, where only a few parameters need to be suitably adjusted. As a consequence

of this approach, a local modulation of a semiconductor, e.g., of a heterostructure, can be

well described in terms of a change of these parameters within the energy range of interest.

As a result, it is possible to introduce a much simpler Schr¨odinger equation, which only

depends on these parameters.

In optics, the simplest model describing the propagation of waves in a periodic dielectric

is provided by the case of a distributed Bragg reﬂector. Here, a modulation with period

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摘要：

Fastdispersiontailoringofmulti-modephotoniccrystalresonatorsFrancescoRinaldoTalenti1;2;,StefanWabnitz1;3,InesGhorbel2,SylvainCombrie2,LucaAimone-Giggio2,AlfredoDeRossi21DipartimentodiIngegneriadell'Informazione,ElettronicaeTelecomunicazioni,SapienzaUniversityofRome,00184Rome,Italy.2ThalesResearch...

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Fast dispersion tailoring of multi-mode photonic crystal resonators Francesco Rinaldo Talenti12 Stefan Wabnitz13 In es Ghorbel2

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