HAL Id: hal-00581731
https://hal.archives-ouvertes.fr/hal-00581731
Submitted on 18 Apr 2021
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Distributed under a Creative Commons Attribution| 4.0 International License
On the generation of octave-spanning optical frequency
combs using monolithic whispering-gallery-mode
microresonators
Y.K. Chembo, N. Yu
To cite this version:
Y.K. Chembo, N. Yu. On the generation of octave-spanning optical frequency combs using mono-
lithic whispering-gallery-mode microresonators. Optics Letters, Optical Society of America - OSA
Publishing, 2010, 35 (16), pp.2696-2698. �10.1364/OL.35.002696�. �hal-00581731�
On the generation of octave-spanning optical
frequency combs using monolithic
whispering-gallery-mode micro resonators
Yanne K. Chembo
1,2,
* and Nan Yu
1
1
Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California 91109, USA
2
Current address: Optics Department, FEMTO-ST Institute (UMR CNRS 6174), 16 Route de Gray, 25030 Besançon, France
*Corresponding author: yanne.chembo@jpl.nasa.gov
Octave-spanning optical frequency combs are especially interesting in optical metrology owing to the ability of self-
referencing. We report a theoretical study on the generation of octave-spanning combs in the whispering gallery
modes of a microresonator. Through a modal expansion model simulation in a calcium fluoride microcavity,
we show that a combination of suitable pump power, Kerr nonlinearity, and dispersion profile can lead to stable
and robust octave-spanning optical frequency combs.
Optical frequency combs are sets of equidistant and ex-
tremely narrow spectral lines in the UV, visible, or IR
ranges [
1–3]. They are typically generated using ultrafast
mode-locked lasers in combination with highly nonlinear
photonic crystal fibers. Recently it has been shown that
they can also be generated using high-Q whispering-
gallery-mode (WGM) resonators [
4–6]. In this case,
photons are resonantly injected into the resonator and
strongly confined into the toruslike WGMs. The ultrahigh
Q-factor leads to long photon storage times, which en-
hances the Kerr nonlinearity of the microresonator bulk
medium. Various cavity eigenmodes can, therefore, be
excited through four-wave mixing (FWM) as the photons
sequentially cascade from the pump to the other WGMs.
In comparison with their mode-locked lasers counter-
parts, WGM optical frequency comb generators are char-
acterized by a significantly reduced size and power
consumption, along with a high repetition rate. They
are, therefore, particularly interesting to miniaturization,
chip integrat ion, and space applications.
For purposes of self-referencing in optical metrology,
it is important that the comb spans over at least one
octave, as it allows a phase-coherent link between the
optical carrier frequency and the comb spacing rf [
1–3].
While there were several experimental demonstrations of
comb generation, there is only one reported observation
of an octave-spanning comb [
7]. At the theoretical level,
the comprehensive stud y is difficult for two main rea-
sons: first, there is no standard model for the comb
dynamics in the literature, and second, numerical simu-
lations over an octave spectral span of several hundreds
of nanometers are computationally challenging. Interest-
ingly, Agha et al. have developed a theoretical model of
comb generation in microcavities based on the nonlinear
Schrödinger equation [
8], and they were able to generate
a comb of a 100 nm wavelength span. In this Letter, we
report the successful simulatio n of an octave-spanning
optical frequency comb using a modal expansion ap-
proach. Our results strongly suggest that experimental
generation of these combs can be obtained in similar
WGM resonators.
We consider a calcium fluoride (CaF
2
) microdisk ob-
tained from a truncated microsphere of radius a ¼ 35 μm
(FSR ¼ 1 THz; FSR, free spectral range), as displayed in
Fig.
1. The small radius of the disk is chosen mainly out of
consideration of reasonable computational feasibility.
The orthonormal eigenmodes of the spherical resonator
V depend on the degenerated angular eigennumber ℓ and
on the polarization p (TE or TM). The eigenmodes ϒ
ℓp
ðrÞ
of the cavity can be explicitly written as ϒ
ℓ;TE
ðrÞ ≃
iϒ
ℓ;TE
ðr; θ; ϕÞe
θ
and ϒ
ℓ;TM
ðrÞ ≃ ϒ
ℓ;TM
ðr; θ; ϕÞe
r
, with
ϒ
ℓp
ðr; θ; ϕÞ¼
ð−1Þ
ℓ
ℓ
1
4
2
1
2
π
3
4
S
ℓp
ðrÞe
−
1
2
ℓðθ−
π
2
Þ
2
e
iℓϕ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R
þ∞
0
S
2
ℓp
ðrÞr
2
dr
q
; ð1Þ
where S
ℓp
is the radial profile of the WGMs [
9], while e
r
, e
θ
,
and e
ϕ
are the orthonormal vectors in spherical coordi-
nates. The electric field is expanded as Eðr;tÞ¼
P
μ
1
2
E
μ
ðtÞe
iω
μ
t
ϒ
μ
ðrÞþc:c:, where c.c. stands for the com-
plex conjugate, while μ ≡ fℓ;pg stands for the modes of
slowly varying amplitude E
μ
ðtÞ and frequency ω
μ
. These
eigenfrequencies are approximated by [
10]
ω
ℓp
¼
c
nðω
ℓp
Þa
ℓ þ
1
2
þ ξ
1
ℓ þ
1
2
2
1
3
−
p
ffiffiffiffiffiffiffiffiffiffiffiffiffi
n
2
0
− 1
q
þ
3
20
ξ
2
1
ℓ þ
1
2
2
−
1
3
þ O
ℓ þ
1
2
−
2
3
; ð2Þ
where p is equal to n
0
≡ nðω
0
Þ for a TE polarization and
1=n
0
for TM, while ξ
1
¼ 2:338 is the first root of the Airy
Fig. 1. (Color online) Lateral view of the microdisk resonator,
and two-dimensional representation of the WGM ℓ
0
¼ 192 (TE).
1
function Aið−zÞ. These eigenfrequencies are weakly
unequidistant because of the material and geometrical
dispersions. One should note that the polar dependence
of the eigenmodes is exp½−
1
2
ℓðθ −
π
2
Þ
2
, so the angular con-
finement of a WGM is of the order of 1=
ffiffi
ℓ
p
. As a conse-
quence, the solutions of the microsphere are still valid
for the truncated microdisk of Fig.
1 as long as the thick-
ness d of the microdisk is larger than the angular confine-
ment. The wavelength λ
0
¼ 2πc=Ω
0
of the pump laser is
finely tuned around 1556:8 nm, corresponding to the
TE mode of order ℓ
0
¼ 192. The loaded quality factor of
this resonantly coupled mode is set to Q
0
¼ 10
8
.
We introduce the normalized electric field A
μ
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ε
0
n
2
0
=2ℏω
μ
q
E
μ
, where jA
μ
j
2
is the instantaneous photon
number in the mode μ, and n
0
is the real part of the re-
fractive index at the laser frequency Ω
0
. The total electric
field Eðr;tÞ obeys the Maxwell wave equation, with a
complex, field- and frequency-dependent permittivity.
After He rmitian inner-product projection onto the
WGMs, the spatiotemporal wave equation is reduced
to the following finite set of coupled rate equations:
_
A
η
¼ −
1
2
Δω
η
A
η
þ
1
2
Δω
η
δ
η0
F
0
e
iσt
− ig
0
X
α;β;μ
Λ
αβμ
η
A
α
A
β
A
μ
e
iϖ
αβμη
t
: ð3Þ
In this equation, Δω
η
¼ 2Υ
η
ω
η
n
a
=n
0
is the modal band-
width, where n
a
is the imaginary part of the refraction
index responsible for material absorption, and Γ
η
¼
R
V
∥ϒ
η
∥
2
dV is the modal confinement factor. The FWM
gain at the eigenfrequency ω
0
of the mode ℓ
0
is g
0
¼
n
2
cℏω
2
0
=n
2
0
V
0
, where n
2
is the nonlinear Kerr coefficient,
and V
0
¼½
R
V
∥ϒ
0
∥
4
dV
−1
is the effective volume of the
mode ℓ
0
. The intermodal coupling factor
Λ
αβμ
η
¼
ω
2
μ
ω
2
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ω
α
ω
β
ω
μ
ω
3
η
s
R
V
½ϒ
η
· ϒ
μ
½ϒ
β
· ϒ
α
dV
R
V
∥ϒ
0
∥
4
dV
ð4Þ
defines the coupling strength between the four interact-
ing modes α, β , μ, and η, which depend on their power
density overlap. As indicated by the Kronecker symbol
δ
η0
, the external field F
0
is only resonant with the mode
ℓ
0
, with a detuning σ ¼ Ω
0
− ω
0
. Th e ideal resonance con-
dition occurs when the modal FWM frequency detuning
ϖ
αβμη
¼ ω
α
− ω
β
þ ω
μ
− ω
η
vanishes. It corresponds to
the FWM interactions ℏω
α
þ ℏω
μ
→ ℏω
β
þ ℏω
η
for
which the energy and the total angular mome ntum of
the interacting photons are conserved (ℓ
α
þ ℓ
μ
¼ ℓ
β
þℓ
η
). In a dispersionless cavity, the eigenmodes are
perfectly equidistant and ϖ
αβμη
¼ 0. We consider the case
where the microresonator is pumped at resonance
(σ ¼ 0). The threshold number of intracavity photons
needed for comb generation is equal to jA
th
j
2
¼
Δω
0
=2g
0
¼ n
2
0
V
0
=2ℏω
0
n
2
cQ, where Δω
0
is the band-
width of the mode ℓ
0
.
We have simulated Eq. (
3) numerically and studied the
resulting comb spectrum. Once the geometry, the bulk
medium, and the losses of the microresonator are given,
the only free parameters of the system are the laser pump
power jF
0
j
2
and its detuning frequency σ relative to the
WGM resonance. In particular, material dispersion is
fixed by the frequency-dependent refraction index nðωÞ.
To characterize the cavity dispersion [
11], we use the ma-
terial dispersion values tabulated by the Sellmeier expan-
sion of [
12]. The geometrical dispe rsion is determined by
Eq. (
2). The rate Eqs. (3) have been numerically inte-
grated using the fourth-order Runge–Kutta algorithm.
The cavity is pumped at the WGM of order ℓ
0
¼ 192.We
included 201 modes ranging from ℓ
min
¼ 92 to ℓ
max
¼ 292.
The initial condition is vacuum ground states defined by
hjA
ℓ
ð0Þj
2
i¼
1
2
, where the brackets stand for ensemble
average over ℓ.
Figure
2 displays two sample results of the numerical
simulation. One of the most important parameters is the
total photon number jA
in
j
2
inside the cavity. When the
intracavity power is set 3 times above the threshold,
the comb already spans beyond one octave [Fig.
2(a)].
The spectrum is, however, noticeably irregular and domi-
nated by sparsely located spec tral components. This
strong spectral modulation indicates that the hyperpa-
rametric interactions are dominated by degenerate FWM.
The degenerate interaction only generates pairwise
comb lines that satisfy the required stability conditions.
Nondegenerate FWM is responsible for filling the inter-
stability regions and creating a more uniform comb line
distribution [
13]. In Fig. 2(b), the intracavity power is in-
creased to 4 times the threshold value. The spectrum is
now more uniform, thereby indicating that nondegene-
rate FWM interactions are much more strongly excited.
We have also observed in our simulations that comb gen-
eration depends on the frequency detuning σ and on the
pump power.
In the simulation study, we also found that the use of
strong pumping in this nonlinear system may lead to
irregular oscillations in the temporal domain. Such oscil-
lations lead to the emergence of parasitic modulation
sidebands in the comb spectra (still within the bandwidth
of the WGMs) [
13]. For wide comb span generation,
Fig. 2. (Color online) Numerical simulation of an octave-
spanning optical frequency comb. The intracavity power values
are (a) jA
in
j
2
¼ 3jA
th
j
2
and (b) jA
in
j
2
¼ 4jA
th
j
2
.
2
therefore, one should have sufficient power to excite as
many WGMs as possible, but not too much as to avoid
these temporal instabilities. It is also noteworthy to point
out that the simulation is computationally intensive. In
the resonator, there is a huge number of FWM interac-
tions of the kind Λ
αβμ
η
A
α
A
β
A
μ
e
iϖ
αβμη
t
. All of them have
to be considered simultaneously. For example, when
we simulate N modes (201 in our case), there are ap-
proximately
4
3
N
2
∼ 5 × 10
6
interactions.
In conclusion, we have carried out a simulation study
on octave-spanning optical frequency comb generation
with monolithic WGM microresonators. Our results show
that, from a dynamical point of view, and as far as the
dispersion and nonlinearity are concerned, it is possible
to generate octave-spanning combs in WGM resonators.
The choice of a small resonator was mainly driven by the
consideration of the numbers of modes needed in the si-
mulation and the corresponding complexity of it. Our re-
sults suggest that octave-spanning combs should be
possible in resonators of smaller FSRs. However, smaller
FSRs mean more comb lines in a given span. A comb of
FSR in tens of gigahertz would already contain several
thousand WGMs. It is expected that the required pump
power will increase accordingly. It is still an open ques-
tion whether the higher pump power will induce tempor-
al instability before the octave span is achieved in this
case. It is also well known that self-phase modulation lo-
cally modifies the refraction index and, thereby, the
mode profiles of WGMs. This effect is generally referred
to as Kerr lensing. It could play an important role in
comb generation when the microcavity is strongly
pumped abo ve threshold. Temperature stability may be
another important technical issue. In fact, the strong con-
finement of WGMs in small volumes generates thermal
lensing that affects the various WGM characteristics
and, hence, the comb generation. Future work should
also consider stochastic thermal fluctuations with em-
phasis on their effects on the phase noise performances
of the comb lines.
This work was performed at the Jet Propulsion Labora-
tory (JPL), California Institute of Technology, under a
contract with NASA. Yanne K. Chembo acknowledges
a fellowship from the NASA Postdoctoral Program, admi-
nistered by Oak Ridge Associated Universities. Authors
also acknowledge logistic support from the JPL Super-
computing and Visualization Facility.
References
1. S. T. Cundiff and J. Ye, Rev. Mod. Phys. 75 , 325 (2003).
2. J. L. Hall, Rev. Mod. Phys. 78, 1279 (2006).
3. T. W. Hänsch, Rev. Mod. Phys. 78, 1297 (2006).
4. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R.
Holzwarth, and T. J. Kippenberg, Nature 450, 1214 (2007).
5. A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I.
Solomatine, D. Seidel, and L. Maleki, Phys. Rev. Lett.
101, 093902 (2008).
6. I. S. Grudinin, N. Yu, and L. Maleki, Opt. Lett. 34, 878 (2009).
7. P. Del’Haye, T. Herr, E. Gavartin, R. Holzwarth, and T. J.
Kippenberg, “Octave spanning frequency comb on a chip,”
arXiv:0912.4890v1 (2009).
8. I. H. Agha, Y. Okawachi, and A. L. Gaeta, Opt. Express 17,
16209 (2009).
9. B. R. Johnson, J. Opt. Soc. Am. A 10, 343 (1993).
10. S. Schiller, Appl. Opt. 32, 2181 (1993).
11. P. Del’Haye, O. Arcizet, M. L. Gorodetsky, R. Holzwarth,
and T. J. Kippenberg, Nat. Photon. 3, 529 (2009).
12. M. Daimon and A. Masumura, Appl. Opt. 41, 5275 (2002).
13. Y. K. Chembo, D. V. Strekalov, and N. Yu, Phys. Rev. Lett.
104, 103902 (2010).
3
