Bandwidth and Conversion-Eciency Analysis of Kerr Soliton Combs in Dual-Pumped Resonators with Anomalous Dispersion E. Gasmi1H. Peng2C. Koos2and W. Reichel1

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Bandwidth and Conversion-Eﬃciency Analysis of Kerr Soliton Combs

in Dual-Pumped Resonators with Anomalous Dispersion

E. Gasmi,1H. Peng,2C. Koos,2and W. Reichel1, ∗

1Institute for Analysis (IANA), Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany

2Institute of Photonics and Quantum Electronics (IPQ),

Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany

Kerr frequency combs generated in high-Q microresonators oﬀer an immense potential in many

applications, and predicting and quantifying their behavior, performance and stability is key to

systematic device design. Based on an extension of the Lugiato-Lefever equation we investigate in

this paper the perspectives of changing the pump scheme from the well-understood monochromatic

pump to a dual-tone conﬁguration simultaneously pumping two modes. For the case of anomalous

dispersion we give a detailed study of the optimal choices of detuning oﬀsets and division of total

pump power between the two modes in order to optimize single-soliton comb states with respect

to performance metrics like power conversion eﬃciency and bandwidth. Our approach allows also

to quantify the performance metrics of the optimal single-soliton comb states and determine their

trends over a wide range of technically relevant parameters.

I. INTRODUCTION AND MAIN RESULTS

Optical frequency combs have revolutionized many appli-

cations, comprising optical frequency metrology [1], spec-

troscopy [2, 3], optical frequency synthesizer [4, 5], optical

atomic clocks [6], ultrafast optical ranging [7], and high-

capacity optical communications using massively parallel

wavelength-division multiplexing (WDM) [8]. The re-

cent and rapid development of chip-scale Kerr soliton

comb generators oﬀers the prospects of realizing highly

integrated devices which oﬀer compactness, portability,

and robustness, while being amenable to mass production

and featuring low power consumption [9]. Whereas Kerr

soliton combs have conventionally been generated by us-

ing a monochromatic pump, dual-tone pumping conﬁg-

urations permit to achieve threshold-less comb genera-

tion in both normal and anomalous dispersion regimes

[10, 11], while stabilizing the comb-tone spacing to a

well-deﬁned frequency [12, 13]. The dual mode pumping

scheme can be implemented either by using a phase- or

intensity-modulated continuous-wave laser or two lasers

with diﬀerent wavelengths. Prior works theoretically in-

vestigated the dynamical properties of dissipative cavity

soliton generation in a dual-mode-pumped Kerr microres-

onator by using the Lugiato-Lefever equation (LLE) with

the addition of a secondary pump term [14]. However,

a comprehensive study of the optimal pumping condi-

tions for attaining the broadest comb bandwidth and the

highest power conversion eﬃciency in the anomalous dis-

persion regime is still lacking.

In this paper we study a variant of the LLE based on a

modiﬁcation for dual-tone pumping [15], and we use this

equation for a more detailed study of the beneﬁts of dual-

tone pumping. Focussing on resonators with anomalous

dispersion, we ﬁnd that dual-tone pumping allows to sig-

niﬁcantly improve key performance metrics of Kerr fre-

∗wolfgang.reichel@kit.edu

quency combs such as bandwidth and power conversion

eﬃciency. Mathematically, Kerr comb dynamics with a

single pumped mode have been described by the LLE, a

damped, driven and detuned nonlinear Schr¨odinger equa-

tion [16–18]. Our modiﬁcation of the LLE arises due to

a forcing term which describes the pumping of two res-

onator modes instead of only a single one.

Using this equation as a base, we exploit numerical

path continuation methods for a more detailed analysis of

comb properties, the results of which can be summarized

as follows:

(1) We show that pumping two modes is advantageous

to pumping only one mode.

(2) We present heuristic insights for ﬁnding the optimal

detuning parameters that provide the most local-

ized single-soliton states.

(3) We determined the optimal power distribution be-

tween the two pumped modes, which corresponds

to a symmetric distribution where 50% of the power

is pumped into each mode1. This power distribu-

tion simultaneously optimizes all performance met-

rics (comb bandwidth, full-width at half-maximum

in time domain, and power conversion eﬃciency)

in case equal detuning oﬀsets between pump tones

and nearest resonant modes are used.

(4) Under optimal power distribution we determined

trends of the performance metrics w.r.t. varying

dispersion and normalized total pump power.

This paper is organized as follows: In Section II we

introduce the Lugiato-Lefever model for a dual-pumped

1For purposes of simplifying the analysis this was exactly the case

discussed by the authors in [10]. Our ﬁndings validate their as-

sumption of the pumps having equal amplitude and phase de-

tuning.

arXiv:2210.09760v1 [physics.optics] 18 Oct 2022

ring resonator. In Section III we present the main ideas

for ﬁnding localized solitons in the case of pumping two

adjacent modes. Section IV is dedicated to the deter-

mination of the optimal power distribution between the

two pumped modes. Here we use the comb bandwidth,

the power conversion eﬃciency and the full-width at half-

maximum as performance metrics. In Section V we pro-

vide trends for varying dispersion/forcing of this per-

formance metrics under the provision of optimal equal

power distribution between the two pumped modes. In

Section VI we describe the optimal solitons achieved by

pumping two arbitrarily distanced modes. Appendix A is

dedicated to the derivation of the Lugiato-Lefever model

for a dual-pumped ring resonator. In Appendix B we

explain the details of the heuristic algorithm for ﬁnding

localized solitons in the case of pumping two adjacent

modes and Appendix C contains the heuristic for the

case of pumping two arbitrarily distanced modes.

II. LUGIATO-LEFEVER MODEL FOR A

DUAL-PUMPED RING RESONATOR

Kerr comb dynamics are described by the LLE, a

damped, driven and detuned nonlinear Schr¨odinger equa-

tion [16–18]. As in [15] we use a variant of the LLE

modiﬁed for two-mode pumping, for which we provide a

derivation of equation (1) starting from a system of non-

linear coupled mode equations in physical quantities in

Appendix A. Using dimensionless, normalized quantities,

this equation takes the form

ı∂a

∂τ =−da00 −(ı −ζ0)a−|a|2a+ ıf0+ ıf1eı(k1x−ν1τ).(1)

Here, a(τ, x) is 2π-periodic in xand represents the op-

tical intracavity ﬁeld as a function of normalized time

τ=κt/2 and angular position x∈[0,2π] within the ring

resonator. The constant κ > 0 describes the cavity decay

rate and d= 2d2/κ > 0 quantiﬁes the anomalous dis-

persion in the system (2d2corresponds to the diﬀerence

between two neighboring FSRs at the center frequency

ω0). Since the numbering k∈Zof the resonant modes

in the cavity is relative to the ﬁrst pumped mode k0= 0

we denote with k1∈Nthe second pumped mode (there

is no loss of generality to take k1as a positive integer

since k1and −k1are symmetric modes). Since there are

now two pumped modes there will also be two normal-

ized detuning parameters denoted by ζ0= 2(ω0−ωp0)/κ

and ζ1= 2(ωk1−ωp1)/κ. They describe the oﬀsets of

the input pump frequencies ωp0and ωp1to the clos-

est resonance frequency ω0and ωk1of the microres-

onator, respectively. Finally f0, f1represent the normal-

ized power of the input pumps. If we set ∆ζ=ζ0−ζ1

and ν1= ∆ζ+dk2

1then (after several transformations,

cf. Appendix A) equation (1) emerges with the speciﬁc

form of the second pump f1eı(k1x−ν1τ).

In the case f1= 0, equation (1) amounts to the case of

pumping only one mode. This case has been thoroughly

studied, e.g. in [17–26]. In this paper we are interested

in the case f16= 0. The particular form of the pump

term ıf0+ ıf1eı(k1x−ν1τ)suggests to perform a change

of variables into a moving coordinate s=x−ωτ with

ω=ν1/k1and study solutions of (1) of the form a(τ, x) =

u(x−ωτ). These traveling-wave solutions propagate with

speed ωin the resonator, and their proﬁle usolves the

stationary ordinary diﬀerential equation

−du00 + ıωu0−(ı −ζ0)u−|u|2u+ ıf0+ ıf1eık1s= 0,(2)

where uis again 2π-periodic in s. In Fourier modes aand

uare represented as a(τ, x) = Pk∈Zˆak(τ)eıkx,u(s) =

Pk∈Zˆukeıks. The intracavity power Pof the ﬁeld aat

time τis given by

P=X

k∈Z|ˆak(τ)|2=1

2πZ2π

0|a(τ, x)|2dx.

Since the Fourier modes of aand uare related by ˆak(τ) =

ˆuke−ıkωτ one ﬁnds P=Pk∈Z|ˆuk|2=1

2πR2π

0|u(s)|2ds.

In particular, Pis independent2of the time, and since

R2π

0|u|2ds = Re R2π

0(f0+f1eık1s)¯u ds we see that P≤

f2:=f2

0+f2

1, i.e., the intracavity power cannot exceed

the normalized total input power. Details are given at

the end of Appendix A. Here, the notation ¯zdenotes the

complex conjugate of the complex number z∈C.

III. HEURISTIC FOR FINDING LOCALIZED

SOLITONS IN THE CASE OF PUMPING TWO

ADJACENT MODES

In the following section, we explain the main idea of the

heuristic for ﬁnding strongly localized solutions of (2),

where two adjacent modes are pumped, i.e. the pumped

modes are k0= 0 and k1= 1. In Appendix B we provide

a more detailed explanation, and in Appendix C we show

how the heuristic can be adapted to arbitrary values of

k1∈N. The parameters d > 0, k1= 1, f0and f1are

ﬁxed, and our goal is to ﬁnd optimally localized solutions

by varying the parameters ζ0and ωsince they can be

inﬂuenced by the choice of the pump frequencies ωp0and

ωp1through the relations

ζ0=2

κω0−ωp0, ω =2

κω0−ωp0−(ω1−ωp1) + d2.

Optimality is understood as minimality with respect to

the full-width at half-maximum (FWHM) of the ﬁeld dis-

tribution |u|2in the time domain. We have developed our

heuristic by using the Matlab package pde2path (cf. [27],

2In fact, the power |ˆuk|2=|ˆak(τ)|2in each mode is independent

of time.

[28]) which has been designed to numerically treat con-

tinuation and bifurcation in boundary value problems for

systems of PDEs.3

In short, the basic algorithm is explained as follows:

First we obtain a single-peak solution for the correct

value of the parameter f1(ignoring the values of the

parameters ζ0and ω). Then we alternately run a con-

tinuation algorithm by varying either the ζ0- or the ω-

parameter (while keeping the other parameter ﬁxed) and

detect among the continued solutions the soliton uwith

minimal FWHM of |u|2in the time domain. We denote

the soliton obtained from the j-th ζ0-optimization as Aj

and the one obtained from the j-th ω-optimization as Bj.

We stop the algorithm when the relative change of the

FWHM of Bj+1 and Bjis suﬃciently small. In our nu-

merical experiments it was always suﬃcient to perform

at most three optimizations in both of the variables ζ0

and ω.

In Fig. 1(a)-(c) we plotted the spatial power distribu-

tions of the solitons Ajand Bjfor two iteration steps

j= 1,2 and three diﬀerent choices of the parameters d,

fand f1. It is well visible that the solitons get more lo-

calized after every optimization step and that the solitons

A2and B2from the second iteration steps do not diﬀer

signiﬁcantly. In the second column of Fig. 1 in (b) and

(e) the blue soliton A2is not visible, since it is covered by

the almost identical magenta soliton B2. In the second

row Fig. 1(d)-(f) we show the spectral power distribu-

tions. The ﬁnal magenta comb B2covers almost entirely

the blue comb A2. The third row of Fig. 1 contains infor-

mation on the spectral stability of the optimized solitons.

This will be explained next.

Stability of optimal solitons. To investigate the stabil-

ity of the solitons, we use the transformation a(τ, x) =

b(τ, x −ωτ) to rewrite (1) as

∂b

∂τ =−ı−db00 +ıωb0−(ı−ζ0)b−|b|2b+ıf0+ıf1eık1s,(3)

where bis again 2π-periodic in s. Solutions uof (2) corre-

spond to stationary solutions b(τ, s) = u(s) of (3). Spec-

tral stability is based on the following considerations. Let

b(τ, s)≈u(s) + φ(s)eλτ +ψ(s)e¯

λτ with 2π-periodic func-

tions φ, ψ, and insert this ansatz into (3). After keeping

only the linear terms in φand ψ, we ﬁnd that φ, ψ have

to satisfy the eigenvalue equation

Lφ

ψ=λφ

ψ

with the linearized operator

L= ıdd2

ds2+ωd

ds −1−ıζ+ 2ı|u|2ıu2

−ı¯u2−ıdd2

ds2+ωd

ds −1+ıζ−2ı|u|2!.

We see that the perturbation φ(s)eλτ +ψ(s)e¯

λτ will tend

to zero if and only if the eigenvalues λof Llie in the

left complex plane. Using this criterion, we found that

the optimized solitons (optimized w.r.t. ζ0and ωby the

above heuristic) discussed in this section are all spec-

trally stable. To show this, we computed the eigenvalues

of the ﬁnite-element discretization of the operator Land

observed that they entirely belong to the left complex

plane, cf. Fig. 1(g)-(i). One sees that there is always

an eigenvalue very close to 0. The reason for this is the

following. The optimized solitons are found near turn-

ing points along branches of the ζ0-continuation, cf. Ap-

pendix B. These turning points are necessarily associated

with a 0 eigenvalue of the linearized operator L. Hence,

3Continuation and bifurcation solvers for boundary value prob-

lems (on which pde2path is based) allow to globally study the

variety of diﬀerent stationary comb states by exploiting the full

range of technically available parameters. In contrast, time-

integration solvers mostly only allow to access speciﬁc comb

states which strongly depend on the chosen device parameters

and initial conditions.

for ubeing in the vicinity of a turning point, there will

be an eigenvalue of Lvery close to 0.

IV. OPTIMAL POWER DISTRIBUTION WHEN

PUMPING TWO ADJACENT MODES

In this section we answer the question which amount of

the normalized total input power f2=f2

0+f2

1needs to

be pumped into each mode in order to obtain the best

soliton, i.e., we determine the optimal power distribu-

tion between the two pumped modes. The power dis-

tribution is described as (f0, f1) = (fcos ϕ, f sin ϕ) with

ϕ∈[0,2π). As before, we assume anomalous dispersion

d > 0 and ﬁx the indices k0= 0 and k1= 1 of the two

pumped modes. Additionally, the normalized total input

power f2is given. Armed with the heuristic from Sec-

tion III we are able to identify for any ﬁxed ϕ∈[0,2π) a

1-soliton with the strongest spatial localization, i.e., with

minimal FWHM.

Using this approach, we calculate for each such a comb

state u(s) = Pk∈Zˆukeıks the power conversion eﬃciency

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BandwidthandConversion-EciencyAnalysisofKerrSolitonCombsinDual-PumpedResonatorswithAnomalousDispersionE.Gasmi,1H.Peng,2C.Koos,2andW.Reichel1,1InstituteforAnalysis(IANA),KarlsruheInstituteofTechnology,76131Karlsruhe,Germany2InstituteofPhotonicsandQuantumElectronics(IPQ),KarlsruheInstituteofTechnolo...

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Bandwidth and Conversion-Eciency Analysis of Kerr Soliton Combs in Dual-Pumped Resonators with Anomalous Dispersion E. Gasmi1H. Peng2C. Koos2and W. Reichel1

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