Spatial-mode-interaction-induced dispersive waves
and their active tuning in microresonators
Q
I
-F
AN
Y
ANG
,
†
X
U
Y
I
,
†
K
I
Y
OUL
Y
ANG
,
AND
K
ERRY
V
AHALA
*
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
*Corresponding author: vahala@caltech.edu
Received 3 June 2016; revised 16 September 2016; accepted 16 September 2016 (Doc. ID 267672); published 5 October 2016
The nonlinear propagation of optical pulses in dielectric wave-
guides and resonators induces a wide range of remarkable inter-
actions. One example is dispersive-wave generation, the optical
analog of Cherenkov radiation. These waves play an essential
role in the fiber-optic spectral broadeners used in spectroscopy
and metrology. Dispersive waves form when a soliton pulse
begins to radiate power as a result of higher-order dispersion.
Recently, dispersive-wave generation in microcavities has been
reported by phase matching the waves to dissipative Kerr
solitons. Here, it is shown that spatial mode interactions within
a microcavity can be used to induce dispersive waves. The
soliton self-frequency shift is also shown to enable fine tuning
control of the dispersive-wave frequency. Both this mechanism
and spatial mode interactions allow spectral control of these
important waves in microresonators.
© 2016 Optical Society
of America
OCIS codes:
(190.5530) Pulse propagation and temporal solitons;
(140.3945) Microcavities; (190.0190) Nonlinear optics.
http://dx.doi.org/10.1364/OPTICA.3.001132
If the spectrum of a soliton pulse extends into regions where
second-order dispersion changes sign, then radiation into a new
pulse, the dispersive wave, may occur at a phase-matching wave-
length [
1
,
2
]. The generation of these waves is analogous to
Cherenkov radiation [
3
] and extends the spectral reach of optical
pulses [
4
]. The recent ability to control dispersion in microreso-
nators 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 fre-
quency microcombs [
6
] and octave-spanning double-dispersive
waves [
7
]. Dispersive-wave generation in optical fibers has tradi-
tionally relied upon control of geometrical dispersion in conjunc-
tion with the intrinsic material dispersion of the dielectric [
4
], and
this same method has been successfully demonstrated in micro-
resonators [
5
]. Recently, spatial mode interactions in multimode
fiber have also been used for this purpose [
8
–
10
].
Here, spatial mode interactions within a microresonator are
used to phase match a soliton pulse to a dispersive wave. These
mode interactions often frustrate the formation of solitons [
11
]
and, as a result, microresonators are typically designed to mini-
mize or exclude entirely the resulting modal avoided crossings
[
5
,
12
–
15
]. Also, while dispersive wave phase matching is nor-
mally induced by more gradual variations in dispersion, spatial-
mode interactions produce spectrally abrupt variations that can
activate a dispersive wave in the vicinity of a narrowband soliton.
Below, the demonstration of dispersive-wave generation by this
process is presented after characterizing two strongly interacting
spatial-mode families. The phase matching of the dispersive wave
to the soliton is then studied, including 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)
[
5
,
13
,
16
–
18
]. It is shown that this mechanism enables active
tuning control of the dispersive wave by pump tuning.
In the experiment, an ultrahigh-
Q
silica microresonator
(3 mm diameter) with a 22 GHz free-spectral range (FSR) was
prepared [
19
]. Typical intrinsic quality factors were in excess of
200 million (cavity linewidths were less than 1 MHz). Mode
dispersion was characterized from 183.92 THz (1630 nm) to
199.86 THz (1500 nm) by fiber-taper coupling to a tunable
external-cavity diode laser and calibrating the laser frequency
scan using a Mach
–
Zehnder interferometer [
13
]. Multiple mode
families were observed, and their measured frequency spectra
are plotted versus mode number,
μ
, as the blue points in
Fig.
1(a)
. In the plot, a linear dispersion term corresponding
to the FSR of the soliton-forming mode family (
Δ
ω
−
) at mode
number zero is subtracted so that a
relative-mode-frequency
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 relative mode
frequencies.
The mode frequencies can be accurately modeled using a
coupled-mode approach. Accordingly, consider two mode fam-
ilies (
A
and
B
) that initially do not interact and that feature
frequencies
ω
A;B
μ
. An interaction between the mode families
is introduced that is characterized by a coupling rate
G
. The
coupling produces two hybrid mode families with upper/
lower-branch mode frequencies
ω
μ
given by the following
expression [
20
–
22
]:
Letter
Vol. 3, No. 10 / October 2016 /
Optica
1132
2334-2536/16/101132-04 Journal © 2016 Optical Society of America
ω
μ
ω
A
μ
ω
B
μ
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
G
2
ω
A
μ
−
ω
B
μ
2
∕
4
q
:
(1)
Note that, in the limit of
j
ω
A
μ
−
ω
B
μ
j
≫
G
, the frequencies
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 dispersion 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.
1(a)
. For mode family
A
:
D
1
A
∕
2
π
21.9733 GHz
,
D
2
A
∕
2
π
15.2 kHz
,and
D
3
A
∕
2
π
−
14.7 Hz
.Formode
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 in Eq. (
1
), relative-mode frequencies for the
hybrid mode families (
Δ
ω
μ
≡
ω
μ
−
ω
0
−
μ
D
1
A
where
ω
0
≡
ω
A
0
) are plotted in Fig.
1(a)
(green) and show good agree-
ment with the measurements (blue points). An improved fitting is
possible using a least squares approach in Eq. (
1
). As an aside, the
two mode families (
Δ
ω
) were observed to couple nearly equally
to the tapered fiber with approximately 20% transmission.
The soliton studied here is a dissipative Kerr soliton (DKS).
The formation of DKSs has recently been described in both
optical fiber [
23
] and microresonators [
5
,
12
–
15
]. This type of
dissipative soliton [
24
] forms through a double balance of second-
order dispersion with the Kerr nonlinearity and cavity loss with
Kerr-induced parametric gain [
12
]. The optical spectrum of a
DKS pumped at
μ
0
(193.45 THz or 1549.7 nm) using a fiber
laser is presented in Fig.
1(b)
. The soliton is triggered and stabi-
lized using the method described in [
13
,
25
]. For comparison, the
ideal
Sech
2
spectral profile that would occur under conditions of
pure second-order dispersion [
5
,
12
,
13
] [mode
A
, dashed orange
curve in Fig.
1(a)
] is provided as the red envelope in Fig.
1(b)
.A
small SSFS [
5
,
13
,
16
–
18
] is apparent in the measured soliton
spectrum, as indicated by the spectral displacement of the soliton
spectral center relative to the pumping frequency. The perturba-
tions to the ideal spectral 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 feature is
apparent (maximum near 198.62 THz or
μ
235
). In contrast
to the weak-avoided-crossing-induced distortion for
μ
<
0
, the
dispersive wave results from a resonance condition (see discussion
below) and the comb teeth are accordingly enhanced in strength.
The coherence of the soliton and dispersive wave is verified by
measuring the electrical spectrum of the detected soliton and dis-
persive-wave pulse train using a photodetector [inset of Fig.
1(b)
].
Phase matching between the soliton and the dispersive 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 repetition frequency) is resonant
with the
μ
th frequency of the soliton-forming mode family,
i.e.,
ω
p
ω
r
μ
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
graphical interpretation of the phase-matching condition based
on the relative mode frequency of Fig.
1(a)
,
ω
0
D
1
A
μ
is sub-
tracted from both sides of the phase-matching condition to give
the following condition:
Δ
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.
If the soliton repetition frequency equals the FSR at
μ
0
(i.e.,
ω
r
D
1
A
), then the r.h.s. of Eq. (
2
) is the horizontal dashed
black line in Fig.
1(a)
[repeated in Fig.
2(a)
]. Under these circum-
stances, the dispersive wave phase matches to the soliton pulse at
the crossing of that line with the soliton-forming mode branch.
However, while the mode dispersion profile
Δ
ω
−
μ
is deter-
mined 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 maxi-
mum. Defining this offset as
Ω
, the repetition frequency is given
by the following equation [
26
,
27
]:
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 SSFS
[
5
,
13
,
16
–
18
]. In this work,
Ω
is dominated by the Raman inter-
action, because the typical dispersive-wave power is
<
0.2%
of the
soliton power, causing a negligible dispersive-wave recoil (recoil
of less than one mode). Photothermal-induced change in
D
1
A
is another possible contribution that will vary
ω
r
as pumping
Fig. 1.
Dispersive-wave generation by spatial mode interaction.
(a) Measured relative mode 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 Eq. (
1
) are
shown in green, and the unperturbed mode families are shown in orange.
The dashed horizontal black line determines phase matching for
ω
r
D
1
A
. (b) Measured soliton optical spectrum with dispersive-wave
feature is shown. For comparison, a
Sech
2
fitting is provided in red. The
pump frequency (black) and soliton center frequency (green) indicate a
Raman-induced SSFS [also see Fig.
2(c)
]. A microwave beatnote of the
photodetected soliton and dispersive wave is shown in the inset (fre-
quency scale is offset by 21.973 GHz; resolution bandwidth is 10 kHz).
Letter
Vol. 3, No. 10 / October 2016 /
Optica
1133
is varied [
28
]. However, the thermal tuning of
D
1
A
is estimated to
be
∼
−
4.5 kHz
∕
mW
(by measurement of resonant frequency
photothermal shift of
∼
−
40 MHz
∕
mW
). With total soliton
power of less than 1 mW [
13
], this photothermal-induced change
in repetition frequency is negligible compared with that caused
by the Raman self-frequency shift (see below).
Combining Eqs. (
2
) and (
3
) gives the following phase-
matching condition:
Δ
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 increasing soliton power (and
bandwidth), the plot of the r.h.s. of Eq. (
4
) versus
μ
acquires
an increasingly negative slope [green dashed line in Fig.
2(a)
].
The phase-matching mode number,
μ
μ
DW
, therefore also
increases (i.e., the dispersive wave shifts to a higher optical
frequency) with increasing soliton power. The two soliton spectra
presented in Fig.
2(b)
illustrate this effect (red spectrum is lower
power and has the lower dispersive-wave frequency). Figure
2(c)
(right) also shows a series of higher-resoluton scans of the disper-
sive 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 dis-
persive-wave frequency were measured for a series of soliton
powers that were set by controlling the cavity-pump detuning
frequency (
δω
) using the method in Refs. [
13
,
25
].
ω
r
was mea-
sured using an electrical spectrum analyzer after photodetection of
the resonator optical output. The offset frequency
Ω
was mea-
sured on an optical spectrum analyzer by fitting the center of
the optical spectrum [see Fig.
1(b)
] to determine the spectral
maximum and then measuring the wavelength offset relative to
the pump. This same spectral fitting also allows determination
of the soliton pulse width,
τ
s
[
13
]. Once the soliton pulsewidth
is known, the pump
–
resonator frequency detuning operating
point can be inferred using
δω
≈
D
2
∕
2
D
2
1
τ
2
s
[
13
].
δω
∕
2
π
ranged
between 7.8 and 21.1 MHz during the measurement. As an
aside, a plot of
Ω
versus
1
∕
τ
4
s
in Fig.
2(c)
(left) verifies that
Ω
is dominated by the Raman self-frequency shift [
18
].
The soliton repetition rate is plotted versus
Ω
in Fig.
2(d)
and
is fitted using Eq. (
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.
1(a)
]. The dispersive-wave frequency is also plotted
in Fig.
2(d)
versus
Ω
and compared with a calculation using
Eq. (
4
). In this calculation,
Δ
ω
−
μ
is approximated using a
linear expansion in
μ
near
μ
200
. Also, a
−
60 kHz
offset is
added to
D
1
A
∕
2
π
in
Δ
ω
−
μ
due to the calibration uncertainty
(
∼
100 kHz
) of FSR [
29
]. No other free parameters are used in
the plot.
Spatial mode interactions provide a way to phase match a DKS
to a dispersive wave. The degree to which these interactions can
be engineered and controlled is an active area of investigation.
Fig. 2.
Dispersive-wave phase-matching condition and Raman-induced frequency shift. (a) Relative mode frequencies for the soliton and interaction
mode families are shown [see Fig.
1(a)
] with dispersive-wave phase matching as dashed lines [see Eq. (
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
(arrows). (b) Soliton optical spectra corresponding to small (red) and large (blue) cavity
–
laser detuning (
δω
).
Sech
2
fitting of the spectral envelope is shown
as the orange curves. (c) Left: SSFS,
Ω
, versus
1
∕
τ
4
s
(
τ
s
is pulse width). The theoretical line is calculated with
Q
166 million
(measured) and Raman
shock time 2.7 fs [
18
]. Right: dispersive-wave spectra with cavity
–
laser detuning (soliton power and bandwidth) increasing from lower to upper trace.
(d) Measured dispersive-wave peak frequencies (red points) and soliton repetition rate (blue points) are plotted versus SSFS. The dashed blue line i
s a plot
of Eq. (
3
). The dashed red line uses Eq. (
4
) to determine the dispersive-wave frequency (
≈
μ
DW
D
1
A
ω
0
) as described in the text. The offset for the
repetition rate vertical scale is
D
1
A
21.9733 GHz
.
Letter
Vol. 3, No. 10 / October 2016 /
Optica
1134
Geometrical control of dispersion over broad spectral spans using
microfabrication methods [
30
] could be applicable for dispersive-
wave control.
It has been shown theoretically and through measurement
that the dispersive-wave frequency can be actively tuned be-
cause of coupling of the soliton repetition rate to the offset
frequency
Ω
. In the silica microcavities tested here, this offset
is dominated by Raman-induced SSFS, and the dispersive wave
is predicted and observed to tune to higher frequencies with in-
creasing soliton power and bandwidth. As a further test of the
theory, the dependence of repetition frequency on SSFS was com-
bined with measurement to extract resonator dispersion param-
eters, which compared well with direct measurements based on
resonator dispersion characterization. The dispersion induced by
modal interactions in the tested device has been measured and
accurately modeled using a coupled-mode formalism. While the
limitations imposed upon the spectral proximity of the dispersive
wave to the soliton are under investigation, it seems possible that
more complex resonator designs could not only engineer the
placement of these crossings, but also locate multiple avoided
crossings near a soliton so as to induce multiplets of disper-
sive waves.
Note: During submission of this work, Matsko
et al.
, reported
on Cherenkov radiation by avoided mode crossings in microre-
sonators [
31
].
Funding.
Defense Advanced Research Projects Agency
(DARPA); National Aeronautics and Space Administration
(NASA); Kavli Nanoscience Institute.
†
These authors contributed equally to this work.
REFERENCES
1. P. Wai, C. R. Menyuk, Y. Lee, and H. Chen, Opt. Lett.
11
, 464 (1986).
2. N. Akhmediev and M. Karlsson, Phys. Rev. A
51
, 2602 (1995).
3. P. A. Cherenkov, Dokl. Akad. Nauk SSSR
2
, 451 (1934).
4. J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys.
78
, 1135 (2006).
5. V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. Pfeiffer, M.
Gorodetsky, and T. Kippenberg, Science
351
, 357 (2016).
6. V. Brasch, E. Lucas, J. D. Jost, M. Geiselmann, and T. J. Kippenberg,
“
Self-referencing of an on-chip soliton Kerr frequency comb without
external broadening,
”
arXiv: 1605.02801 (2016).
7. Q. Li, T. C. Briles, D. Westly, J. Stone, R. Ilic, S. Diddams, S. Papp, and
K. Srinivasan, in
Frontiers in Optics
(Optical Society of America, 2015),
paper FW6C.5.
8. J. Cheng, M. E. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüner-
Nielsen, and D. Jakobsen, Opt. Lett.
37
, 4410 (2012).
9. G. Manili, A. Tonello, D. Modotto, M. Andreana, V. Couderc, U. Minoni,
and S. Wabnitz, Opt. Lett.
37
, 4101 (2012).
10. D. Modotto, M. Andreana, K. Krupa, G. Manili, U. Minoni, A. Tonello,
V. Couderc, A. Barthélémy, A. Labruyère, B. M. Shalaby, P. Leproux,
S. Wabnitz, and A. B. Aceves, J. Opt. Soc. Am. B
32
, 1676 (2015).
11. T. Herr, V. Brasch, J. Jost, I. Mirgorodskiy, G. Lihachev, M. Gorodetsky,
and T. Kippenberg, Phys. Rev. Lett.
113
, 123901 (2014).
12. T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky, and
T. Kippenberg, Nat. Photonics
8
, 145 (2013).
13. X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, Optica
2
, 1078
(2015).
14. P.-H. Wang, J. A. Jaramillo-Villegas, Y. Xuan, X. Xue, C. Bao, D. E.
Leaird, M. Qi, and A. M. Weiner, Opt. Express
24
, 10890 (2016).
15. C. Joshi, J. K. Jang, K. Luke, X. Ji, S. A. Miller, A. Klenner, Y. Okawachi,
M. Lipson, and A. L. Gaeta, Opt. Lett.
41
, 2565 (2016).
16. C. Milián, A. V. Gorbach, M. Taki, A. V. Yulin, and D. V. Skryabin,
Phys. Rev. A
92
, 033851 (2015).
17. M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. Pfeiffer, M. Zervas,
M. Geiselmann, and T. J. Kippenberg, Phys. Rev. Lett.
116
, 103902
(2016).
18. X. Yi, Q.-F. Yang, K. Y. Yang, and K. Vahala, Opt. Lett.
41
, 3419 (2016).
19. H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala,
Nat. Photonics
6
, 369 (2012).
20. H. A. Haus and W. P. Huang, Proc. IEEE
79
, 1505 (1991).
21. J. Wiersig, Phys. Rev. Lett.
97
, 253901 (2006).
22. Y. Liu, Y. Xuan, X. Xue, P.-H. Wang, S. Chen, A. J. Metcalf, J. Wang,
D. E. Leaird, M. Qi, and A. M. Weiner, Optica
1
, 137 (2014).
23. F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M. Haelterman,
Nat. Photonics
4
, 471 (2010).
24. A. Ankiewicz and N. Akhmediev,
Dissipative Solitons: From Optics to
Biology and Medicine
(Springer, 2008).
25. X. Yi, Q.-F. Yang, K. Y. Yang, and K. Vahala, Opt. Lett.
41
, 2037 (2016).
26. A. B. Matsko and L. Maleki, Opt. Express
21
, 28862 (2013).
27. J. K. Jang, M. Erkintalo, S. Coen, and S. G. Murdoch, Nat. Commun.
6
,
7370 (2015).
28. P. Del
’
Haye, O. Arcizet, A. Schliesser, R. Holzwarth, and T. J.
Kippenberg, Phys. Rev. Lett.
101
, 053903 (2008).
29. J. Li, H. Lee, K. Y. Yang, and K. J. Vahala, Opt. Express
20
, 26337
(2012).
30. K. Y. Yang, K. Beha, D. C. Cole, X. Yi, P. Del
’
Haye, H. Lee, J. Li, D. Y.
Oh, S. A. Diddams, S. B. Papp, and K. J. Vahala, Nat. Photonics
10
, 316
(2016).
31. A. B. Matsko, W. Liang, A. A. Savchenkov, D. Eliyahu, and L. Maleki,
Opt. Lett.
41
, 2907 (2016).
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Vol. 3, No. 10 / October 2016 /
Optica
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