arXiv:1606.00954v1 [physics.optics] 3 Jun 2016
Spatial-mode-interaction-induced dispersive-waves and
their active tuning in
microresonators
Qi-Fan Yang
∗
, Xu Yi
∗
, Ki Youl Yang and Kerry Vahala
†
T. J. Watson Laboratory of Applied Physics, California Institute of
Technology, Pasadena, California 91125, USA.
∗
These authors contributed equally to this work.
†
Corresponding author: vahala@caltech.edu
The nonlinear propagation of optical pulses in di-
electric waveguides and resonators provides a lab-
oratory to investigate a wide range of remarkable
interactions. Many of the resulting phenomena
find applications in optical systems. One exam-
ple is dispersive wave generation, the optical ana-
log of Cherenkov radiation. These waves have
an essential role in fiber spectral broadeners that
are routinely used in spectrocopy and metrology.
Dispersive waves form when a soliton pulse begins
to radiate power as a result of higher-order dis-
persion. Recently, dispersive wave generation in
microcavities has been reported by phase match-
ing the waves to dissipative Kerr cavity (DKC)
solitons. Here, it is shown that spatial mode in-
teractions within a microcavity can also be used
to induce dispersive waves. These interactions
are normally avoided altogether in DKC soliton
generation. The soliton self frequency shift is also
shown to induce fine tuning control of the disper-
sive wave frequency. Both this mechanism and
spatial mode interactions provide a new method
to spectrally control these important waves.
If the spectrum of a soliton pulse extends into regions
where second-order dispersion changes sign, then radia-
tion into a new pulse, the dispersive wave, may occur at
a phase matching wavelength
1,2
. The generation of these
waves is analogous to Cherenkov radiation
3
and greatly
extends the spectral reach of optical pulses
4
. The re-
cent ability to control dispersion in microresonators has
allowed accurate spectral placement of dispersive waves
relative to a radiating cavity soliton
5
. Such dispersion-
engineered control has made possible 2f-3f self referencing
of frequency microcombs
6
and octave-spanning double-
dispersive waves
7
. To date, dispersive wave generation in
both resonators and in optical fiber has resulted from ma-
nipulation of geometrical dispersion in conjunction with
the intrinsic material dispersion of the dielectric
4,5
.
Here, a different mechanism for dispersive wave gen-
eration is demonstrated: spatial mode interaction within
a microresonator. These mode interactions often frus-
trate the formation of DKC solitons
8
and, as a result,
resonators are typically designed to minimize or exclude
entirely the resulting modal avoided crossing
5,9–12
. Also,
while dispersive-wave phase matching is normally in-
duced by more gradual variations in dispersion, spatial
mode interactions produce spectrally abrupt variations
that can activate a dispersive wave in the vicinity of a
narrow-band soliton. These interactions can, in princi-
ple, occur mutliple times, suggesting the possibility of
dispersive wave multiplets scattered from a single soliton.
Below, the demonstration of dispersive wave generation
is presented after characterizing two strongly interacting
spatial mode families. The phase matching of this wave
is then studied including for the first time the effect of
soliton frequency offset relative to the pump as is caused
by soliton recoil or by the Raman-induced soliton self-
frequency shift (SSFS). It is shown that this mechanism
enables active tuning control of the dispersive wave by
pump tuning.
In a resonator, the formation of dissipative Kerr cavity
(DKC) solitons induces phase locking of the longitudinal
modes belonging to a specific spatial mode family of the
resonator
9,13
. In the absence of third-order dispersion, a
soliton will form by balance of the second-order disper-
sion term with the Kerr nonlinearity. The introduction
third-order dispersion will perturb the soliton and can
lead to radiation of a dispersive wave
5
. To form solitons
in microcavities interactions between the soliton mode
family and other spatial mode families are eliminated or
minimized so as to ensure that second-order dispersion
is dominant
8
. Here, these interactions are intentionally
used to enable dispersive wave generation.
In the experiment, an ultra-high-Q silica micro-
resonator (3 mm diameter) with a 22 GHz free-spectral-
range (FSR) was prepared
14
. Typical intrinsic quality
factors were in excess of 300 million (cavity linewidths
were less than 1MHz). Mode dispersion was character-
ized from 183.92 THz (1630 nm) to 199.86 THz (1500 nm)
by fiber-taper coupling a tunable external-cavity diode
laser (ECDL) and calibrating the frequency scan using
a Mach-Zehnder interferometer (MZI)
10
. Multiple mode
families were observed and their measured frequency dis-
persion spectra are presented as the blue points in fig.
1(a). In the plot, a linear dispersion term correspond-
ing to the FSR of the soliton-forming mode family at
mode number zero is subtracted so that a
relative
-
mode
-
f requency
is plotted. Mode zero is by convention the
mode that is optically pumped to form the soliton. Three
weak perturbations of the soliton mode family dispersion
are observed for
μ <
0. The mode family associated with
one of the perturbations is plotted as the nearly vertical
line of blue points. A much stronger interaction occurs
near
μ
= 165 causing a strong avoided mode crossing
that redirects the soliton-forming branch to lower rela-
tive mode frequencies.
2
Measurement
unperturbed fit
hybrid mode fit
Pump
184
186
188
190
192
194
196
198
200
202
Power (20 dB/div)
frequency (THz)
-400
-200
0
200
400
0
500
1000
1500
-400
-200
0
200
400
(ω−ω
0
−μ
D
1A
)/2
π
(MHz)
Mode number (
μ
)
Soliton
center
~sech
2
mode B
∆ω
−
mode A
(a)
(b)
FIG. 1: Dispersive wave generation by spatial mode
interaction. (a) Measured relative frequencies (blue
points) of the soliton-forming mode family and the
interaction mode family. Mode number
μ
= 0
corresponds to the pump laser frequency of 193.45 THz
(1549.7 nm). Hybrid mode frequencies calculated from
eqn.(1) are shown in green and the unperturbed mode
families are shown in orange. The dashed black line
gives the phase matching condition in cases where the
soliton repetition rate equals the microcavity FSR at
μ
= 0. (b) Typical DKC soliton optical spectrum with
dispersive wave feature. For comparison, a
Sech
2
fitting
is shown in red. The pump frequency (black) and
soliton center frequency (green) indicate a
Raman-induced soliton self-frequency shift.
The dispersion of the two interacting mode families
can be accurately modeled using a coupled mode ap-
proach. Accordingly, consider two mode families (
A
and
B
) that initially do not interact and that feature fre-
quency dispersion spectra
ω
A,B
(
μ
). An interaction be-
tween the mode families is introduced that is character-
ized by a coupling rate
G
. The coupling produces two
hybrid mode families with upper/lower-branch mode fre-
quencies
ω
±
(
μ
) given by the following expression
15–17
,
ω
±
(
μ
) =
ω
A
(
μ
) +
ω
B
(
μ
)
2
±
√
G
2
+ [
ω
A
(
μ
)
−
ω
B
(
μ
)]
2
/
4
(1)
Note that in the limit of
|
ω
A
(
μ
)
−
ω
B
(
μ
)
| ≫
G
, the fre-
quencies
ω
±
approach the frequencies,
ω
A,B
(
μ
), of the
non-interacting mode families. The form of
ω
A,B
(
μ
)
are determined using this fact by fitting them within
the regions
μ <
50 and
μ >
280 of the measured dis-
persion spectra to the following equation:
ω
A,B
(
μ
) =
ω
A,B
(0) +
D
1
A,B
μ
+
D
2
,A,B
μ
2
/
2 +
D
3
,A,B
μ
3
/
6, which is
a third-order Taylor expansion of each mode family about
mode number
μ
= 0. The corresponding fits are shown as
the dashed orange curves in fig. 1a. For mode family
A
:
D
1
A
/
2
π
= 21
.
9733 GHz,
D
2
A
/
2
π
= 15
.
2 kHz,
D
3
A
/
2
π
=
−
14
.
7 Hz; and for mode family
B
:
D
1
B
/
2
π
= 21
.
9654
GHz,
D
2
B
/
2
π
= 18
.
6 kHz,
D
3
B
/
2
π
=
−
17
.
2 Hz and
ω
B
(0)
−
ω
A
(0) = 1
.
261 GHz. The coupling coefficient,
G
, is determined by the minimum frequency difference of
two branches and gives
G/
2
π
=106.5 MHz. Using these
parameters, hybrid mode frequencies
ω
±
(
μ
) (eqn. (1))
are plotted in green in fig. 1a and show good agreement
with the measured dispersion branches. An improved
fitting is possible by applying a least squares approach
using eqn. (1). As an aside, the definition of the relative
frequency for fig. 1a (and fig. 2a) is
ω
0
≡
ω
A
(0).
The optical spectrum of a DKC soliton pumped at
μ
= 0 (193.45 THz or 1549.7 nm) using a fiber laser
is presented in fig. 1b. The soliton is triggered and sta-
bilized using the method described elsewhere
10,18
. For
comparison, the ideal hyperbolic
Sech
2
spectral profile
that would occur under conditions of pure second-order
dispersion
5,9,10
(mode A dashed orange curve in fig. 1a)
is provided as the red envelope in fig. 1b. A small soli-
ton self-frequency shift (SSFS)
5,10,19
is apparent in the
measured soliton spectrum as indicated by the spectral
displacement of the soliton spectral center relative to the
pumping frequency. The perturbations to the ideal spec-
tral envelope that are caused by both the weak modal
crossings (
μ <
0) as well as the strong avoided modal
crossing are apparent. For
μ >
0 a dispersive wave fea-
ture (maximum near 198.62 THz or
μ
= 235) is apparent.
The phase matching condition for this dispersive wave is
considered next.
Phase matching between the soliton and the disper-
sive wave occurs when the
μ
th
soliton line at
ω
p
+
ω
r
μ
(
ω
p
is the pump frequency and
ω
r
is the soliton repeti-
tion frequency) is resonant with the
μ
th
frequency of the
soliton-forming mode family, i.e.,
ω
−
(
μ
) =
ω
p
+
ω
r
μ
(the
soliton forms on the lower frequency branch,
ω
−
(
μ
)). As
an aside, the Kerr shift for mode
μ
is much smaller than
other terms in this analysis and is neglected in the phase
matching condition. So that it is possible to use a graph-
ical interpretation of the phase matching condition based
on the relative frequency of fig. 1a,
ω
0
+
D
1
A
μ
is sub-
tracted from both sides of the phase matching condition
to give the following condition,
∆
ω
−
(
μ
) = (
ω
r
−
D
1
A
)
μ
−
δω
(2)
where
δω
≡
ω
0
−
ω
P
is the detuning of the resonator
relative to the pump frequency, and where ∆
ω
−
(
μ
)
≡
ω
−
(
μ
)
−
ω
0
−
D
1
A
μ
is the soliton forming branch plotted
in fig. 1a.
If the soliton repetition frequency equals the FSR at
μ
= 0 (i.e.,
ω
r
=
D
1
A
), then the r.h.s. of eqn. (2)
3
Relative Power (100 dB/div)
-500
-600
-400
-300
-200
-100
0
dispersive wave peak (THz)
Pump
188
190
192
194
196
198
-60
-40
-20
Power (dBm)
Optical frequency (THz)
197.5
198.0
198.5
Optical frequency (THz)
-200
-100
0
100
200
0
0
-100
-200
-300
500
Mode number (
μ
)
250
-250
Mode freq
Comb freq w/o SSFS
Comb freq w/ SSFS
Repetition Rate - offset (kHz)
198.4
198.2
198.0
198.6
Measurement
Linear fit
(a)
(b)
(d)
(c)
(ω−ω
0
−μ
D
1A
)/2
π
(MHz)
∆ω
−
Self-frequency shift
Ω
/2
π
(GHz)
FIG. 2: Dispersive wave phase matching condition and Raman-induce
d frequency shift. (a) Soliton and interaction
mode family dispersion curves are shown (see fig. 1a) with phase mat
ching dashed lines (eqn. (4)). The black line is
the case where
ω
r
=
D
1
A
and the green line includes a Raman-induced change in
ω
r
. The intersection of the soliton
branch with these lines is the dispersive wave phase matching point (a
rrows). (b) Soliton optical spectra
corresponding to small (red) and large (blue) cavity-laser detunin
g (
δω
).
Sech
2
fitting of the spectrum envelope is
shown as the orange curves. (c) High resoluton scan of dispersive
wave spectra with cavity-laser detuning (soliton
power and bandwidth) increasing from lower to upper trace. (d) Dis
persive wave peak frequencies (orange points)
and soliton repetition rate (blue points) are plotted versus soliton s
elf-frequency shift. The linear fit (dash line)
agree with the model. The offset for the repetition rate vertical sc
ale is =
D
1
A
= 21
.
9733 GHz.
is the horizontal dashed black line in fig. 1a (repeated
in fig. 2a). Under these circumstances the dispersive
wave phase matches to the soliton pulse at the cross-
ing of that line with the soliton-forming mode branch.
However, while the mode dispersion profile (∆
ω
−
(
μ
)) is
determined entirely by the resonator geometry and the
dielectric material properties, the soliton repetition rate
ω
r
depends upon frequency offsets between the pump
and the soliton spectral maximum. Defining this offset
as Ω, the repetition frequency is given by the following
equation
20,21
,
ω
r
=
D
1
A
+
D
2
A
D
1
A
Ω
.
(3)
The offset frequency Ω can be caused by soliton recoil
due to a dispersive wave and also by the Raman-induced
soliton self-frequency shift (SSFS)
5,10,19
. In this work,
Ω is dominated by the Raman interaction, because the
typical dispersive wave power is
<
0
.
2% of the soliton
power and causes a negligible dispersive wave recoil. It is
also noted that photo-thermal-induced change in
D
1
A
is
another possible contribution that will vary
ω
r
as pump-
ing is varied
22
. However, it is estimated to be
∼ −
4
.
5
kHz/mW (by measurement of resonant frequency photo-
thermal shift of
∼ −
40 MHz/mW). With total soliton
power less than 1 mW
10
, this photo-thermal induced
change is therefore also determined to be negligible com-
pared with the effect of the Raman self-frequency-shift
in eqn.(3).
Combining eqns. (2) and (3) gives the following phase
matching condition:
∆
ω
−
(
μ
) =
μ
D
2
A
D
1
A
Ω
−
δω.
(4)
The Raman-induced SSFS is a negative frequency shift
(Ω
<
0) with a magnitude that increases with soliton
bandwidth and average power. Accordingly, with in-
creasing soliton power (and bandwidth), the plot of the
r.h.s. of eqn. (4) versus
μ
acquires an increasingly neg-
ative slope (green dashed line in fig. 2a). The phase
matching mode number,
μ
=
μ
DW
, therefore also in-
creases (i.e., the dispersive wave shifts to a higher optical
frequency) with soliton power. The two soliton spectra
presented in fig. 2b illustrate this effect (red spectrum
is lower power and has the lower dispersive wave fre-
quency). Fig. 2c also shows a series of higher-resoluton
4
scans of the dispersive wave with soliton power increasing
from the lower to upper scans and is, again, consistent
with the prediction.
The frequency shift, Ω, repetition frequency,
ω
r
, and
the dispersive wave frequency were measured for a series
of soliton powers. The soliton power was varied by chang-
ing the cavity-pump detuning frequency (
δω
) using the
method described elsewhere
10,18
.
ω
r
was measured us-
ing an electrical spectrum analyzer after photodetection
of the resonator optical output. The offset frequency
Ω was directly measured on an optical spectrum ana-
lyzer by fitting the center of optical spectrum (see fig.
1b) to determine the spectral maximum and measuring
the wavelength offset relative to the pump. The disper-
sive wave frequency increases with increasing Ω (fig. 2d)
which is again consistent with the graphical interpreta-
tion of eqn. (4). The soliton repetition rate vesus Ω (see
fig. 2d) is fitted using eqn. (3). The intercept closely
agrees with
D
1
A
and the slope allows determination of
D
2
A
/
2
π
= 14
.
7 kHz (in good agreement with 15
.
2 kHz
from fitting to the measured dispersion curve in fig. 1a).
Spatial mode interaction provides a new way to phase
match a DKC soliton to a dispersive wave. The ap-
proach requires a resonator to support at least two spatial
modes. At present, all high-Q systems used for genera-
tion of solitons satisfy this condition. It has been shown
theoretically and through measurement that the disper-
sive wave frequency can be actively tuned because of
coupling to the soliton offset frequency Ω. In the sil-
ica microcavities tested here, this offset is dominated by
the Raman-induced SSFS and the dispersive wave is pre-
dicted and observed to tune to higher frequencies with
increasing soliton power and bandwidth. As a further
test of the theory, the dependence of repetition frequency
on SSFS was combined with measurement to extract res-
onator dispersion parameters, which compared well with
direct measurements based on resonator dispersion char-
acterization. The dispersion induced by modal interac-
tions in the tested device has been measured and accu-
rately modeled using a coupled-mode formalism.
Modal interactions such as those studied here for dis-
persive wave generation can in principle be engineered.
This capability has been demonstrated in silica res-
onators, but with the objective to exclude modal cross-
ings from specific spectral regions so as to enable soli-
ton formation. It seems possible that more complex res-
onator designs could not only engineer the placement of
these crossings, but also locate multiple avoided cross-
ings near a soliton so as to induce multiplets of dispersive
waves.
Note: during submission of this work, Matsko, et. al.,
reported on Cherenkov radiation by avoided mode cross-
ings in microresonators
23
.
FUNDING INFORMATION
The authors gratefully acknowledge the Defense Ad-
vanced Research Projects Agency under the PULSE and
DODOS programs, NASA, the Kavli Nanoscience Insti-
tute.
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