Fourier Synthesis Dispersion Engineering of Photonic Crystal Microrings for Broadband Frequency Combs Gr egory Moille1 2Xiyuan Lu1 2Jordan Stone1 2Daron Westly2and Kartik Srinivasan1 2

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Fourier Synthesis Dispersion Engineering of Photonic Crystal Microrings for

Broadband Frequency Combs

Gr´egory Moille,

1, 2, ∗

Xiyuan Lu,

1, 2

Jordan Stone,

1, 2

Daron Westly,

and Kartik Srinivasan

1, 2

Joint Quantum Institute, NIST/University of Maryland, College Park, USA

Microsystems and Nanotechnology Division, National Institute of Standards and Technology, Gaithersburg, USA

(Dated: July 12, 2023)

Abstract

Dispersion engineering of microring resonators is crucial

for optical frequency comb applications, to achieve tar-

geted bandwidths and powers of individual comb teeth.

However, conventional microrings only present two geo-

metric degrees of freedom – width and thickness – which

limits the degree to which dispersion can be controlled.

We present a technique where we tune individual reso-

nance frequencies for arbitrary dispersion tailoring. Using

a photonic crystal microring resonator that induces cou-

pling to both directions of propagation within the ring,

we investigate an intuitive design based on Fourier syn-

thesis. Here, the desired photonic crystal spatial proﬁle

is obtained through a Fourier relationship with the tar-

geted modal frequency shifts, where each modal shift is

determined based on the corresponding eﬀective index

modulation of the ring. Experimentally, we demonstrate

several distinct dispersion proﬁles over dozens of modes

in transverse magnetic polarization. In contrast, we ﬁnd

that the transverse electric polarization requires a more

advanced model that accounts for the discontinuity of the

ﬁeld at the modulated interface. Finally, we present simula-

tions showing arbitrary frequency comb spectral envelope

tailoring using our Frequency synthesis approach.

Introduction

Frequency combs based on integrated nonlinear microres-

onators are a powerful tool to bring metrology outside

the lab. They allow for low power consumption and porta-

bilty [

] while maintaining metrological quality [

] while

in the dissipative Kerr soliton (DKS) regime. Although

octave-spanning frequency combs – needed for carrier-

envelope stabilization through self-referencing of the comb

– have been demonstrated [

], reaching beyond an octave

is particularly interesting so that the strong pump can be

doubled in self-referencing schemes. Yet, it is extremely

challenging, especially at short wavelengths towards the

visible. Materials that are used for microcomb genera-

tion, including

Si3N4

[

–

AlN

[

], and

LiNbO3

[

present increasingly large normal dispersion the shorter

the wavelength is [

–

]. Modal conﬁnement of the light

in resonators with wavelength-scale cross-sections adds a

geometrical component to the dispersion, which in many

cases is enough to compensate for the normal material

dispersion. However, a simple rectangular cross-section

microresonator does not oﬀer enough degrees of freedom

∗gmoille@umd.edu

to achieve broad enough anomalous dispersion (needed for

bright DKS states) to tackle goals such as spectral band-

widths well-beyond an octave while extending well into

the visible. Alternative approaches have been proposed.

Among them, multi-pumped DKS [

] and pulsed-pump res-

onators [

] have been successful in realizing spectral

bandwidths beyond that of conventional DKS microcombs.

Yet, these solutions often increase the complexity of the

setup required for ﬁeld deployment of these microcombs.

It is thus necessary to create methods through which

one can engineer the dispersion of a microring resonator.

Approaches with multi-layer material stacks [

], com-

plex ring cross-sections [

] and concentrical rings [

] –

each relying on avoided-mode crossings – have generated

much more complex dispersion proﬁles. However, fabrica-

tion has been challenging and may be incompatible with

top-down foundry-like mass fabrication processes [

]. In

addition, broad bandwidth microcombs based on these

approaches have not yet been demonstrated. Nevertheless,

the concept of an avoided-mode crossing, which relies on

mode-coupling, can be harnessed in diﬀerent fashions, for

example, by coupling the clockwise (CW) and counter-

clockwise (CCW) directions of the same transverse optical

mode. Such CW/CCW coupling has been demonstrated

by Lu et al., where modulation of the microresonator

sidewall creates a photonic crystal that frequency splits a

targeted mode without impacting the nearest neighbors,

determined by the number of photonic crystal periods

within the ring circumference [

]. Interestingly, nonlin-

ear states such as optical parametric oscillation [

–

]

and DKSs can be created in this system [

]. Moreover,

this photonic crystal ring concept has been expanded to

modulation amplitudes far beyond a simple perturbation,

where a full band gap of hundreds of gigahertz to several

terahertz is resolved, impacting the band structure among

several neighboring modes [

]. More importantly for this

work, it has been shown that it is possible to introduce

multi-period photonic crystal patternings that create con-

trolled frequency splittings for a few (up to 5) targeted

modes [

]. In particular, sinusoidal ring width modu-

lations corresponding to multiple single modes targeted

simultaneously (i.e. with diﬀerent amplitudes and mod-

ulation periods) can be summed with limited impact to

the other modes. In summary, ref. [

] lays out a method-

ology based on Fourier synthesis for microring dispersion

engineering [

], where a spectral proﬁle of mode coupling

and corresponding shift of several individual modal fre-

quencies follows a discrete Fourier transform of the ring

width modulation.

arXiv:2210.14108v2 [physics.optics] 11 Jul 2023

Mode Number μ Mode Number μ

DW Width

∝D2

Power (dB)

Dint/2π (GHz)

D2/2π (MHz)

H=425 nm

H=405 nm

Ring Width (nm) Mode number M

DW span (µ) P @ DW (dB)

T. (arb. unit)

−100 −50 0 50 100 150

−50

100

Angle

dependant

property

Dint/2π (GHz)

D2/2 µ2

−100 −50 0 50 100 150

−50

−100

800 1050 n-1 nn+1 n+2 n+3

200

-75

anomalous

dispersion

Original

dispersion

Optimized

dispersion

normal

dispersion

(a) (d) (e)

(f)

(b) (c)

Single material layer

+ same thickness=

same fabrication

process

Dispersion engineering limitation of uniform rectangular

cross section microring

Photonic Crystal Microring Fourier Dispersion Engineering

2(n+1)

2(n+2)

Fig. 1 Photonic crystal patterning for broadband microring dispersion engineering. (a) Simulated frequency comb (blue)

assuming the

Dint

proﬁle in red similar to that in ref. [

], and suitable for octave span thanks to dual-DWs. Positive quadratic dispersion

(D2>0) results in the sech2comb envelope near the pump, while further away, higher-order dispersion allows for zero-crossings where

Dint =0. This results in comb teeth on resonance, locally enhancing the power through DWs – and driving the overall bandwidth of the

comb, while D2dictates the width of the sech2envelope. The soliton power (following the sech2envelope) at Dint =0 impacts the

generated DW power. (b) Standard microring resonator where three parameters are available for dispersion engineering, ring radius (

ring width (RW ) and material thickness (H). (c) Variation of D2, power in the DW, and separation between the two potential DW

locations (in units of mode number) as a function of ring width. These plots illustrate the trade-oﬀ between potential microcomb

bandwidth and DW power, where the largest comb bandwidth (bottom panel) requires large D2(top panel) and hence a sharper sech2

envelope that leads to lower available power at the Dint zero crossing for DW enhancement (middle panel). (d) Concept of a photonic

crystal ring where a sinusoidal modulation of the ring at a period 2nwill split the mode M=nthrough CW/CCW coupling, with a

strength dependent on the modulation amplitude. Within a certain regime of modulation amplitude, other azimuthal modes are not

disturbed. Therefore, one can simply add modulations with diﬀerent periods and amplitudes to selectively target diﬀerent modes

with

desired frequency splittings (T= transmission in arbitrary units). (e) The frequency splittings modify Dint , resulting in a higher Dint

branch and a lower Dint branch (shown in green). When a suﬃciently large number of modes are split, the lower Dint branch can be

ﬂattened (i.e., largely reducing the maximum

Dint

value) to bypass the

vs. potential comb bandwidth trade-oﬀ. (f) The resulting ring

resonator has a ring width modulation proﬁle that is the inverse DFT of the splitting of each mode that one wants to design, essentially

equivalent to a Fourier synthesis approach. As a result, many split modes will produce a very localized ring-width modulation. This

introduces challenges that we discuss throughout the rest of the paper.

Yet, whether a vast number of mode-couplings through

Fourier synthesis – which remains a perturbative appraoch

– can be implemented in a predictive manner that yields

precise and eﬃcient dispersion engineering consistent with

the needs of, for example, broadband microresonator fre-

quency combs is still unclear. In this work, we propose to

answer this question with an in-depth study of photonic-

crystal-mediated microring dispersion engineering in the

limit of large (tens of gigahertz) spectral shifts. We demon-

strate that the previously utilized simple analysis that

links the modal frequency shifts directly to ring width

modulation is inaccurate in our regime of interest. In

contrast, modulation of the ring eﬀective refractive index,

which is mapped (nonlinearly) to a ring width modula-

tion, is a better approach. We also show that the po-

larization considered greatly impacts the validity of the

perturbative approach with which this mode-by-mode dis-

persion engineering is predicted using our Fourier synthesis

model. This limit arises from the boundary conditions

on the dominant electric ﬁeld components at discontin-

uous boundaries. Consequently, the transverse magnetic

polarization is more suited for predictive dispersion en-

gineering using our straightforward Fourier synthesis ap-

proach, while the transverse electric polarization may

require an approach based on full three-dimensional nu-

merical simulations of Maxwell’s equations in modulated

microring structures. Using our technique, we fabricate

silicon nitride photonic crystal microrings in which dozens

of resonances are shifted in a controlled fashion by up to

GHz

, compatible with the integrated dispersion mitiga-

tion needed for broadband (e.g., octave-spanning) combs,

and in good agreement with simulations. Finally, we use

coupled Lugiato-Lefever equation modeling to predict the

spectral behavior of microcombs that can be generated

using our dispersion technique, and in particular, the pos-

sibility of considerably extending their bandwidth and the

possibility of creating multi-color Bragg solitons.

Results

Towards a broad and ﬂat microcomb spectrum:

motivating photonic crystal dispersion engineering.

Under the right conditions of pump power and detuning,

microcombs based on the third-order optical nonlinearity

can support DKS states [

]. The usual way of studying

the dynamics of such a system is through the Lugiato-

Lefever equation [

] which is essentially a dissipative

nonlinear Schr¨odinger equation [

] that takes into ac-

count the microring resonator’s periodic boundary con-

ditions while operating under a slow varying envelope

(mean-ﬁeld) approximation [

]. The single DKS solu-

tion in the anomalous dispersion regime follows the well-

known hyperbolic secant (sech) spectral envelope (sech

for spectral intensity). To quantify the resonator disper-

sion, the community usually opts for a Taylor expansion

and deﬁnes a quantity termed the integrated dispersion

Dint =∑k>1

k!µk=ωres −(ω0+D1µ)

, with

being the

linear repetition rate at the pumped mode with resonant

frequency

ω0

, and

the azimuthal mode number refer-

enced to the pumped mode. The higher order dispersion

terms

are of great importance as they drive the shape

of the integrated dispersion and, ultimately, the proper-

ties of the microcomb. The sech

comb envelope width

is inversely proportional to

, which must be positive

for anomalous dispersion. The odd terms

D2k+1

drive the

recoil of the soliton, resulting in the drift of its repeti-

tion rate away from the linear one. The even terms

D2k

are responsible for symmetric zero crossings of

Dint

and

yield dual dispersive waves (DWs) as comb teeth become

resonant at these modal frequencies. The experimental

demonstrations of DWs [

] have fundamentally changed

the landscape of microcombs by bypassing the sech

width

driven by

and expanding it to octave span [

]. In

this work, we will refer to the frequency span between the

Dint =

0 frequencies as the ‘dispersive wave span’ of the

microcomb [Fig. 1(a)-(b)]. Although these DWs have been

the key enabler for broadband microcombs, the power in

these modes relies on the available power in the sech

soliton envelope. Therefore, if

is too large, leading

to a sharp comb envelope close to the pump, the power

available at the DW locations will be insigniﬁcant for

resonant enhancement. This encapsulates the dispersion

engineering challenge for broadband integrated frequency

combs: getting as broad as possible the

Dint

zero-crossings

while keeping the D2>0 portion as ﬂat as possible.

Guided-wave photonics results in wavelength-dependent

light conﬁnement: the longer the wavelength, the larger

the mode and less conﬁned it is. This unique feature

allows one to tailor the dispersion beyond the material

dispersion. For a microring resonator, there are essentially

three user-deﬁned input parameters: ring radius (

ring width (

), and thickness (

) [Fig. 1(b)]. The

mainly acts on

and typically has little inﬂuence on the

higher order dispersion terms. Therefore, most dispersion

engineering eﬀorts focus on

and

. However, with

only these two parameters available, the DW span and

increase together [Fig. 1(c)]. Although increasing the

comb width is the ultimate goal, increasing

reduces

the power available at the

Dint

zero-crossings, ultimately

preventing useful DWs from forming. Thus, with current

dispersion engineering approaches, an apparent trade-oﬀ

exists.

In recent years, a modiﬁed microring resonator has been

developed, where modulation of the ring width allows for

coherent backscattering between the clockwise (CW) and

counter-clockwise (CCW) traveling wave modes [

It has been demonstrated, using a perturbative approach

theoretically and veriﬁed experimentally, that a single

harmonic modulation of the ring width results in a sin-

gle azimuthal mode of a given transverse spatial mode

family experiencing this coupling [

]. These two cou-

pled modes hybridize into symmetric and anti-symmetric

modes, creating a mode splitting proportional to the am-

plitude of the modulation [Fig. 1(d)]. In the unmodulated

ring, the necessity to match the ﬁeld phase after one round

trip results in a set of azimuthal modes described by a

mode number

M∈Z

. When the modulation is applied, a

new spatial period

πRR/M0≪

πRR

becomes important.

Here, the modulation consists of 2

periods around the

ring circumference. In the language of photonic crystals

(PhCs) [

], the frequency splitting created by coupling of

the CW and CCW azimuthal modes at

±M0

is a frequency

band gap, with the Brillouin zone folded at these points

as well. These modulated devices can thus be referred to

as photonic crystal (PhC) microrings, and in the limit of

strong modulation, large band gaps of several terahertz

have been demonstrated while maintaining high optical

quality factor (

) [

]. In addition, smaller modulation

PhC microrings have been used to change the dynam-

ics of DKS formation [

]. For smaller modulations, it

has also been shown that the splitting is linearly propor-

tional to the modulation amplitude employed, and that the

CW/CCW coupling is nearly absent for all modes near the

targeted mode

[

]. Therefore multiple mode splittings

(PhC modulations) can be implemented on a single ring.

Following ref. [

], the design rules then become simple:

a straightforward sum of individual modulation patterns

gives the microring ring width modulation to implement

[Fig. 1(d)], such that

RWmod =∑mA(m)cos (2m(θ+φ))

with

A(m)

the modulation intensity of each targeted mode

and

the microring azimuthal angle. Therefore, the

total ring width modulation in this scheme is, per deﬁni-

tion, the discrete inverse Fourier transformation (DFT)

of the modal coupling envelope A(m).

The frequency shift created by the CW/CCW coupling

can be used to modify the dispersion of the resonator

locally. The integrated dispersion is then shifted by half

the resonance splitting at each coupled mode, creating

two bands. One band is pushed toward higher

Dint

, and

one reduces the value of

Dint

. The latter can overcome the

aforementioned trade-oﬀ in large

and large potential

comb bandwidth if the Fourier synthesis design approach

allows for predictive mode shifts in the tens of gigahertz

range to counterbalance the maximum

Dint

value of an

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FourierSynthesisDispersionEngineeringofPhotonicCrystalMicroringsforBroadbandFrequencyCombsGr´egoryMoille,1,2,∗XiyuanLu,1,2JordanStone,1,2DaronWestly,2andKartikSrinivasan1,21JointQuantumInstitute,NIST/UniversityofMaryland,CollegePark,USA2MicrosystemsandNanotechnologyDivision,NationalInstituteofStanda...

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Fourier Synthesis Dispersion Engineering of Photonic Crystal Microrings for Broadband Frequency Combs Gr egory Moille1 2Xiyuan Lu1 2Jordan Stone1 2Daron Westly2and Kartik Srinivasan1 2

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