Modeling of dual frequency combs and bistable solitons in third-harmonic generation Tobias Hansson12 Pedro Parra-Rivas2 and Stefan Wabnitz2 1Department of Physics Chemistry and Biology

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Modeling of dual frequency combs and bistable solitons in third-harmonic generation

Tobias Hansson1,2,∗, Pedro Parra-Rivas2, and Stefan Wabnitz2

1Department of Physics, Chemistry and Biology,

Link¨oping University, SE-581 83 Link¨oping, Sweden

2Dipartimento di Ingegneria dell’Informazione, Elettronica e Telecomunicazioni,

Sapienza Universit`a di Roma, via Eudossiana 18, 00184 Rome, Italy and

∗Corresponding author: tobias.hansson@liu.se

(Dated: October 19, 2022)

ABSTRACT

Phase-matching of the third-harmonic generation process can be used to extend the emission of

radiation from Kerr microresonators into new spectral regions far from the pump wavelength. Here,

we present a theoretical mean-ﬁeld model for optical frequency combs in a dissipative and nonlinear

χ(3)-based cavity system with parametric coupling between fundamental and third-harmonic waves.

We investigate temporally dispersive dual-comb generation of phase-matched combs with broad

bandwidth, and report conditions for accessing a multistable regime that simultaneously supports

two types of coupled bright cavity solitons. These bistable cavity solitons coexist for the same

pump power and frequency detuning, while featuring dissimilar amplitudes of their individual ﬁeld

components. Third-harmonic generation frequency combs can permit telecom pump laser sources

to simultaneously directly access both the near-infrared and the visible regions, which may be

advantageous for the development of optical clocks and sensing applications.

INTRODUCTION

Optical frequency combs (OFCs) utilize the nonlin-

ear polarization response of a cavity-enclosed dielectric

medium, in order to convert an externally applied pump

ﬁeld to multiple new frequencies. Acting as broadband

and coherent optical sources, OFCs are a key technol-

ogy for enabling a diverse range of applications such as

frequency metrology, optical communications and spec-

troscopy [1–3]. However, conventional Kerr comb syn-

thesizers only emit radiation in a spectral range that is

centered around the pump laser frequency, and generally

require anomalous group-velocity dispersion for the ex-

perimentally accessible formation of phase-locked states

[4,5]. This makes it challenging to form combs in wave-

length ranges which lack suitable pump laser sources, and

in spectral regions that exhibit an eﬀective waveguide

and material dependent normal dispersion.

One way of overcoming these limitations is to exploit

the third-harmonic generation (THG) process of the χ(3)-

nonlinearity, in order to couple pump ﬁeld excitations to

parametric waves at thrice the fundamental frequency

(FF), 3ω1. The THG process is inherent in all transpar-

ent nonlinear media that display a strong Kerr eﬀect, but

in practice it is hampered by the requirement of main-

taining a ﬁxed phase relation, which is necessary for eﬃ-

cient frequency conversion [6]. While many experimental

observations of THG in microcomb devices have relied on

refractive index matching between the fundamental and

higher-order whispering gallery modes, it is also feasible

to accomplish phase-matching through birefringence, pe-

riodic poling and other quasi-phase-matching techniques

[7].

In this work we consider a centrosymmetric nonlin-

ear Kerr resonator system that is engineered to phase-

match the third-harmonic process in order to enable res-

onant dual-comb generation around both the FF and the

third-harmonic (TH) frequency, when the dissipative cav-

ity is driven by a continuous-wave (CW) pump source

at the fundamental frequency ω1. We go beyond pre-

vious studies of cavity THG that have been restricted

to the non-dispersive case with only two interacting fre-

quencies [8,9], by considering the mutual coupling be-

tween sidebands around each carrier wave and the si-

multaneous interaction of all frequency modes. This

system shares similarities with non-phase-matched Kerr

microresonators, that can be modelled by the Lugiato-

Lefever equation (LLE) [10,11] or driven-and-damped

nonlinear Schr¨odinger equation [12]; it is also analogous

to OFCs in quadratically nonlinear resonators, which ex-

ploit cascaded processes, found in χ(2)-nonlinear media

without inversion symmetry, in order to enable coupling

and dual-comb generation around both the FF and the

second-harmonic frequency [13,14]. We note that a sim-

ilar model of THG-assisted four-wave mixing was pub-

lished in Ref. [15] during the ﬁnal preparation of this

manuscript, but with the inclusion of simplifying assump-

tions that limit its applicability to a perturbative THG

regime, and exclude the possibility of generating bistable

cavity solitons.

Previous experimental work has demonstrated the di-

rect emission of visible light by THG from an infrared

pump using both high-Q whispering-gallery-mode and

integrated microresonators [16–20]. The generation of

OFCs by THG acting together with Raman-assisted

spectral broadening in silica based microcavities was also

reported [21]. Theoretical studies of spatially diﬀrac-

tive beam propagation in conservative, cavityless systems

arXiv:2210.09763v1 [physics.optics] 18 Oct 2022

have also shown the possibility of generating both bright

and dark coupled solitary wave structures in the presence

of THG; that exhibit properties such as a power thresh-

old and bistability [22,23]. Given that the dynamics of

our system is governed solely by the fundamental ﬁeld in

the limit of vanishing parametric coupling, one may ex-

pect to ﬁnd a similar homotopic extension of the phase-

locked bright cavity solitons (CSs) that are supported

by the LLE in the case of anomalous group-velocity dis-

persion [24,25]. Surprisingly, we ﬁnd that coupled FF

and TH combs can also support an additional type of CS

with a partially overlapping range of existence. These

dual, two component, solitons share a common trapping

refractive index potential through self- and cross-phase

modulation (SPM/XPM). Moreover, there is the intrigu-

ing possibility that mutual XPM coupling can be used to

overcome the group-velocity mismatch, in order to create

various types of synchronized dual-frequency combs with

locked repetition rates around both the FF and the TH

frequency.

In the following sections, we develop a theoretical

mean-ﬁeld model for a doubly-resonant, cavity-enhanced,

dispersive and nonlinear system, phase-matched for

THG. In particular we show that, as the TH ﬁeld

grows larger, it may not be simply considered as an up-

converted replica of the fundamental comb, but it will

reciprocally inﬂuence the latter through both paramet-

ric coupling and XPM. We investigate the multistability

properties of the homogeneous solution and consider the

importance of modulational instability (MI) in generat-

ing various types of multi-frequency combs. Additionally,

we make a detailed numerical study of the multistable

regime, where we demonstrate the occurrence of bistable

cavity solitons that coexist, when both the FF and the

TH frequency lie in the anomalous dispersion regime.

RESULTS AND DISCUSSION

Model

We consider OFC generation in a dispersive χ(3)-based

resonator system with coupling between fundamental

and third-harmonic ﬁelds, as schematically illustrated in

Fig. 1. The system is assumed to be resonant around

both the driving frequency of the fundamental ﬁeld (FF)

ω1as well as the frequency of the third-harmonic (TH)

ﬁeld ω2= 3ω1. For simplicity, we assume an isotropic

nonlinear polarization response ¯

PNL =0χ(3) ¯

E3and a

linearly polarized electric ﬁeld propagating along the z-

axis, viz.

E=ˆe1

2F1(x, y)A(z, t)ei(k1z−ω1t)+

F2(x, y)B(z, t)ei(k2z−ω2t)+c.c. (1)

where 0is the vacuum permittivity, χ(3) is the third-

order nonlinear susceptibility, ˆeis a unit vector in the

Ain

ω1

Aout

ω1

Bout

ω2= 3ω1

FIG. 1. Schematic of the THG resonator system.

The χ(3)-based nonlinear microresonator is phase-matched for

third-harmonic generation. The resonator is driven by a CW

pump ﬁeld at the fundamental frequency ω1and generates

simultaneous frequency combs around both ω1and ω2= 3ω1.

polarization direction and c.c. denotes complex conju-

gate. The OFC generation dynamics is modelled by

means of a scalar Ikeda map [26] for the evolution of

the temporal ﬁeld during each roundtrip, together with

appropriate boundary conditions for the injection of the

external pump and the coupling of the ﬁelds from one

roundtrip to the next. Starting from Maxwell’s equations

and applying the slowly-varying envelope approximation,

the envelopes of the co-propagating electric ﬁeld of the

fundamental Amand the third-harmonic Bmat the mth

roundtrip are found to obey the following coupled non-

linear equations:

∂Am

∂z =−αc1

2+iD1i∂

∂t Am+iω1n2(ω1)

c×

Q13Bm(A∗

m)2e−i∆kz +Q11|Am|2+ 2Q12|Bm|2Am,

(2)

∂Bm

∂z =−αc2

2+iD2i∂

∂t Bm+iω2n2(ω2)

c×

Q23

3A3

mei∆kz +2Q21|Am|2+Q22|Bm|2Bm,(3)

where zis the longitudinal coordinate and tis time.

The dispersive properties of the medium that are associ-

ated with the non-equidistant resonance spacing are de-

scribed by the Taylor series expansions D1,2(i∂/∂t) =

P∞

n=1(k(n)

1,2/n!)(i∂/∂t)nof the propagation constants

k1,2(ω) with k(n)

1,2=dnk1,2/dωn|ω1,ω2. Here k0

1,2are in-

verse group-velocities, k00

1,2are group-velocity dispersion

coeﬃcients and ∆k= 3k1−k2is a wave-vector mismatch.

Moreover, we have that Qij are modal overlap integrals,

αc1,2are the FF/TH absorption losses, cis the speed of

light in vacuum and n2(ω) = 3χ(3)(ω)/8n(ω) is the non-

linear coeﬃcient, with n(ω) the linear refractive index.

In the case of natural phase-matching we have ∆k= 0

which requires matching of the FF/TH refractive indices

n(ω1) = n(ω2) and implies that ∆2= 3∆1, which is as-

sumed in the following. The transverse overlap integrals

that are needed to account for the variation in spatial

mode proﬁles between families of diﬀerent mode orders

are given by

Q11 =R|F1|4dS

(R|F1|2dS)2, Q12 =R|F1|2|F2|2dS

(R|F1|2dS)(R|F2|2dS),

Q13 =R(F∗

1)3F2dS

(R|F1|2dS)3/2(R|F2|2dS)1/2,

Q21 =R|F1|2|F2|2dS

(R|F1|2dS)(R|F2|2dS), Q22 =R|F2|4dS

(R|F2|2dS)2,

Q23 =RF3

1F∗

2dS

(R|F1|2dS)3/2(R|F2|2dS)1/2,(4)

where Q21 =Q12,Q∗

23 =Q13 and dS =dxdy. These

deﬁnitions reduce to the familiar Kerr coeﬃcient γ=

ω1n2(ω1)/cAeff with Aeff =Q−1

11 in the absence of any

TH ﬁeld.

The ﬁelds at the beginning of the (m+ 1)th roundtrip

are assumed to be related to the ﬁelds at the end of the

mth roundtrip through the boundary conditions

Am+1(0, t) = pθ1Ain +p1−θ1e−iδ1Am(L, t),(5)

Bm+1(0, t) = p1−θ2e−iδ2Bm(L, t),(6)

that model a generic optical coupling, such as the evanes-

cent ﬁeld overlap from a nearby waveguide or tapered

ﬁber, that partially transmits the external pump ﬁeld

Ain while reﬂecting the intracavity ﬁelds from the previ-

ous roundtrip. Here, Lis the length of the cavity, while

θ1,2are the power transmission coeﬃcients and δ1,2are

the phase detunings of the FF/TH ﬁelds from the near-

est cavity resonance. We note that the complementary

case of a down-converting, 3ω1→ω1, optical parametric

oscillator can be modelled by simply moving the pump

term to Eq. (6) for the TH ﬁeld.

The Ikeda map constitutes a complete model for the

temporal and spectral dynamics of a THG cavity based

OFC generation system for general resonance and phase-

matching conditions. But analytical and numerical in-

vestigations can be simpliﬁed in the doubly-resonant case

(θ1,2≈1) by averaging the above map over the roundtrip

length into a mean-ﬁeld model, similar to the LLE. In the

following, we truncate the dispersion to the second-order;

assume the phase-matching to be almost perfect, so that

the coherence length is longer than the cavity length;

and shift to a retarded reference frame moving with the

group-velocity of the driving ﬁeld (k0

1)−1. Following a

derivation, whose details are presented in the Methods

section, we obtain our main system of normalized mean-

ﬁeld evolution equations for the FF and TH ﬁelds Aand

Bas

∂A

∂t =−(1 + i∆1)−iη1

∂2

∂τ 2A+

iκ∗B(A∗)2+ (|A|2+ 2σ|B|2)A+f, (7)

∂B

∂t =−(α+i∆2)−d∂

∂τ −iη2

∂2

∂τ 2B+

i3ρhκ

3A3+ (2σ|A|2+µ|B|2)Bi,(8)

FIG. 2. Multistability of homogeneous solutions. The

plot shows colored parameter regions with 1,3,5 or 7 simulta-

neous homogeneous solutions that coexist for the same nor-

malized detuning ∆1and pump power f.

where tand τare normalized slow- and fast-time vari-

ables, respectively. α=α2/α1is the ratio of the

roundtrip losses, ∆j=δj/α1is the normalized detuning,

d=p2L/(|k00

1|α1)∆k0is the walk-oﬀ parameter that de-

pends on the group-velocity mismatch ∆k0=k0

2−k0

ηj=k00

j/|k00

1|is the ratio of the group-velocity dispersion

coeﬃcients and f=pθ1ω1n2(ω1)LQ11/(cα3

1)Ain is the

normalized input pump ﬁeld. The nonlinear interaction

is governed by the four dimensionless parameters

ρ=n2(ω2)

n2(ω1), µ =Q22

Q11

, σ =Q21

Q11

κ=Q23

Q11

ei∆kL/2sinc(∆kL/2),(9)

that can be assumed to be close to unity, unless the

phase-matching is signiﬁcantly multimodal. It is inter-

esting to note that Eqs. (7-8) are formally similar to mod-

els describing phase-matched doubly-resonant second-

harmonic generation (SHG) in quadratic nonlinear media

with a simultaneous Kerr nonlinearity [14,27]. The two

systems diﬀer mainly in the magnitude of the terms and

in the exponents of the FF that appears in the paramet-

ric coupling terms: these read as BA∗and A2in the case

of second-harmonic generation.

Homogeneous solutions

The response of the system for pump powers below

the threshold for parametric comb generation is charac-

terized by CW emission at both FF and TH frequencies.

We ﬁnd a set of stationary mixed-mode homogeneous so-

lutions by setting the derivatives in Eqs. (7-8) to zero.

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Modelingofdualfrequencycombsandbistablesolitonsinthird-harmonicgenerationTobiasHansson1;2;,PedroParra-Rivas2,andStefanWabnitz21DepartmentofPhysics,ChemistryandBiology,LinkopingUniversity,SE-58183Linkoping,Sweden2DipartimentodiIngegneriadell'Informazione,ElettronicaeTelecomunicazioni,SapienzaUnive...

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Modeling of dual frequency combs and bistable solitons in third-harmonic generation Tobias Hansson12 Pedro Parra-Rivas2 and Stefan Wabnitz2 1Department of Physics Chemistry and Biology

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