of 8
Sideband spectroscopy and dispersion
measurement in microcavities
Jiang Li, Hansuek Lee, Ki Youl Yang, and Kerry J. Vahala
T. J. Watson Laboratory of Applied Physics,
California Institute of Technology, Pasadena, California 91125, USA
vahala@caltech.edu
Abstract:
The measurement of dispersion and its control have become
important considerations in nonlinear devices based on microcavities. A
sideband technique is applied here to accurately measure dispersion in a
microcavity resulting from both geometrical and material contributions.
Moreover, by combining the method with finite element simulations, we
show that mapping of spectral lines to their corresponding transverse mode
families is possible. The method is applicable for high-Q, micro-cavities
having microwave rate free spectral range and has a relative precision of
5
.
5
×
10
6
for a 2 mm disk cavity with FSR of 32.9382 GHz and Q of 150
milllion.
© 2012 Optical Society of America
OCIS codes:
(130.3120) Integrated optics devices; (140.3945) Microcavities; (120.0120) In-
strumentation, measurement, and metrology; (300.6380) Spectroscopy, modulation.
References and links
1. K. J. Vahala, “Optical microcavities,” Nature
424
, 839–846 (2003).
2. A. B. Matkso, and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-Part I: Basics,” IEEE J.
Quantum Electron.
12
, 3–14 (2006).
3. V. S. Ilchenko, and A. B. Matkso, “Optical resonators with whispering-gallery modes-Part II: Applications,”
IEEE J. Quantum Electron.
12
, 15–32 (2006).
4. T. Aoki, B. Dayan, E. Wilcut, W. P. Bowen, A. S. Parkins, T. J. Kippenberg, K. J. Vahala, and H. J. Kimble,
“Observation of strong coupling between one atom and a monolithic microresonator,” Nature
443
, 671–674
(2006).
5. T. J. Kippenberg, and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science
321
, 1172–
1176 (2008).
6. F. Vollmer, and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,”
Nat. Methods
5
, 591–596 (2008).
7. T. Lu, H. Lee, T. Chen, S. Herchak, J. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanopar-
ticle detection using optical microcavities,” Proc. Natl. Acad. Sci. U.S.A.
108
, 5976–5979 (2011).
8. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science
322
, 555–559 (2011).
9. W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Whispering-gallery-mode-
resonator-based ultranarrow linewidth external-cavity semiconductor laser,” Opt. Lett.
35
, 2822–2824 (2010).
10. H. Lee, T. Chen, J. Li, K. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-
resonator on a silicon chip,” Nat. Photon.
6
, 369–373 (2012).
11. J. Li, H. Lee, T. Chen, and K. J. Vahala, “Characterization of a high coherence, Brillouin microcavity laser on
silicon,” Opt. Express
20
, 20170–20180, (2012).
12. T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. Gorodetsky, and T. J. Kip-
penberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photon.
6
, 480–487 (2012).
13. I. Grudinin, A. Matsko, and L. Maleki, “Brillouin lasing with a CaF
2
whispering gallery mode resonator,” Phys.
Rev. Lett.
102
, 043902 (2009).
14. M. Tomes, and T. Carmon, “Photonic micro-electromechanical systems vibrating at X-band (11-GHz) rates,”
Phys. Rev. Lett.
102
, 113601 (2009).
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OSA
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15. V. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-factor and nonlinear properties of optical
whispering-gallery modes,” Phys. Lett. A
137
, 393–397 (1989).
16. D. W. Vernooy, V. S. Ilchenko, H. Mabuchi, E. W. Streed, and H. J. Kimble, “High-Q measurements of fused-
silica microspheres in the near infrared,” Opt. Lett.
23
247–249 (1998).
17. D. K. Armani , T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,”
Nature
421
, 925–928 (2003).
18. A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, and L. Maleki, “Kilohertz optical resonances in dielectric crystal
cavities,” Phys. Rev. A
70
, 051804(R) (2004).
19. S. B. Papp, and S. A. Diddams, “Spectral and temporal characterization of a fused-quartz-microresonator optical
frequency comb,” Phys. Rev. A
84
, 053833 (2011).
20. M. J. Thorpe, R. J. Jones, K. D. Moll, J. Ye, and R. Lalezari, “Precise measurements of optical cavity dispersion
and mirror coating properties via femtosecond combs,” Opt. Express
13
, 882–888 (2005).
21. A. Schliesser, C. Gohle, T. Udem, and T. W. Hansch, “Complete characterization of a broadband high-finesse
cavity using an optical frequency comb,” Opt. Express
14
, 5975–5983 (2006).
22. P. Del’Haye, O. Arcizet, M. L. Gorodetsky, R. Holzwarth, and T. J. Kippenberg, “Frequency comb assisted diode
laser spectroscopy for measurement of microcavity dispersion,” Nat. Photonics
3
, 529–533 (2009).
23. A. A. Savchenkov, E. Rubiola, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Phase noise of whispering gallery
photonic hyper-parametric microwave oscillators,” Opt. Express
16
, 4130–4144 (2008).
24. R. G. DeVoe, C. Fabre, K. Jungmann, J. Hoffnagle, and R. G. Brewer, “Precision optical-frequency-difference
measurements,” Phys. Rev. A
37
, 1802–1805 (1988).
25. J. Li, H. Lee, T. Chen, O. Painter and K. Vahala, “Chip-based Brillouin lasers as spectral purifiers for photonic
systems,” arXiv:1201.4212 (2011).
26. M. Cai, O. Painter, and K. J. Vahala, “Observation of critical coupling in a fiber taper to silica-microsphere
whispering gallery mode system,” Phys. Rev. Lett.
85
, 74–77 (2000).
27. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala, “Ideality in a Fiber-Taper-Coupled Microres-
onator system for application to cavity quantum electrodynamics,” Phys. Rev. Lett.
91
, 043902 (2003).
28. M. Oxborrow, “Traceable 2-d finite-element simulation of the whispering-gallery modes of axisymmetric elec-
tromagnetic resonators,” IEEE Trans. Microw. Theory Tech.
55
, 1209–1218 (2007).
29. O. Arcizet, A. Schliesser, P. DelHaye, R. Holzwarth, and T. J. Kippenberg, “Optical frequency comb generation
in monolithic microresonators,” in
Practical Applications of Microresonators in Optics and Photonics
, ed. A. B.
Matsko, (CRC Press, 2009), Ch. 11.
30. G. Agrawal,
Nonlinear Fiber Optics
(Academic Press, 2001).
1. Introduction
Optical microcavities are widely studied across many areas in photonics [1–3] including topics
in fundamental science such as cavity QED [4] and cavity optomechanics [5], and in appli-
cations such as biosensing [6, 7], microcombs [8] and narrow-linewidth laser sources [9–11].
Beyond Q factor and mode volume, dispersion has become a significant parameter in certain
applications. In the subject of microcombs, dispersion determines the spectral maximum of
the parametric gain, whether comb oscillation initiates on a native mode spacing, and also mi-
crocomb bandwidth at a given pump power [12]. As another example, Brillouin microcavity
lasers require careful matching of free spectral range (FSR) with the Brillouin shift [10,13,14],
and predicting the necessary cavity diameter to achieve this matching requires knowledge of
the cavity dispersion. In addition to measurement of cavity dispersion, a wide range of micro-
cavities (including microspheres [15,16], microtoroids [17], crystalline resonators [18], quartz
micromachined resonartors [19], and wedge resonators [10]) feature many transverse mode
families, and accurate measurement of FSR can provide a way to map these modes with ob-
servable spectral lines.
In microcavities, dispersion has both geometrical (cavity shape) and material contributions,
and, as a result, it depends upon the wavelength and the transverse spatial mode family. The
measurement of FSR as a function of wavelength provides a convenient way to characterize
cavity dispersion. Along these lines, frequency combs have provided a powerful way to mea-
sure dispersion in cavities that can be approximately matched in FSR to the comb repetition
frequency [20, 21]. Also, in cases where this is not possible, a frequency-comb has been used
together with a tunable-diode-laser to measure FSR and dispersion [22]. These techniques,
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while providing fast, accurate and wide band dispersion measurement, require access to a sta-
bilized frequency comb. In another method, “white light”, filtered toa1nmbandwidth, has
been used for dispersion measurement in a CaF
2
whispering gallery microcavity [23]. Multiple
RF beat notes are produced that contain the contribution of all the WGM modes (with different
transverse order and azimuthal order) in the 1 nm bandwidth probed. This is a convenient way
to measure dispersion, but the method does not enable correspondence of the optical spectral
peaks with the RF beat notes. In this letter, we modify a method that has been used to measure
FSR and mirror-induced dispersion in a Fabry-Perot [24] to the case of a microresonator. In
that method two FM sidebands were imposed on an optical carrier and then separately tuned
so as to attain locking with two cavity modes. In the present work we use a single modulation
in combination with a reference interferometer. While not attaining the same level of precision
in the earlier work, the present technique does not require locking to the resonator modes and
provides sufficient precision to measure both the spectral and transverse mode dependence of
dispersion in a high-Q silica microresonator. The technique can be applied in resonators with
FSR in the microwave rate range.
2. Measurement method and uncertainty analysis
A lithium niobate phase modulator and a microwave source are used to generate a sinusoidally,
phase modulated signal on a probe laser at a modulation rate
f
m
close to the FSR of the resonator
such that
FSR
>
f
m
(see Fig. 1(a)). The probe laser is an external cavity diode laser and is
scanned across a cavity resonance so as to produce the oscilloscope trace of the transmitted
power shown in Fig. 1(b). When the phase modulation is “off”, the transmission spectrum of the
cavity has the single Lorentzian line shape as shown in Fig. 1(b) (red curve). However, when the
phase modulation is “on”, the transmission spectrum will show three spectral peaks as the two
side peaks come from the phase modulation sidebands that are coupled to the cavity through the
neighboring resonances of the initial cavity resonance. The offset frequency (
Δ
f
=
FSR
f
m
)is
then measured by using a Mach-Zehnder interferometer (MZI) to create a fringe-like reference
spectrum. If
T
d
and
T
m
are corresponding oscilloscope time intervals for the offset frequency
and neighboring MZI fringe maxima, then the offset frequency is given by
Δ
f
=
T
d
Tm
FSR
M
,
where
FSR
M
is the FSR of the MZI. Upon determination of the offset frequency, the cavity
FSR is determined by
FSR
=
f
m
+
Δ
f
.
Using the above results, the uncertainty of cavity FSR is given by,
δ
FSR
=
δ
f
m
+
T
d
Tm
δ
FSR
M
+
δ
T
d
Tm
FSR
M
+
T
d
δ
T
m
T
2
m
FSR
M
(1)
The first term on the R.H.S of Eq. (1) is the uncertainty of the modulation frequency. It is
determined by frequency uncertainty of the RF synthesizer, which is negligible compared with
the other terms (less than 1Hz).
Concerning the second term in Eq. (1), the FSR of the MZI is measured to kHz level un-
certainty by sending a CW external-cavity diode laser into the MZI and measuring the power
spectral density (PSD) using a balanced photodetector. When the laser frequency is set to the
quadrature point of the MZI, the frequency noise of the laser is discriminated by the MZI
fringes. As shown in [10, 25] the PSD is then proportional to
sinc
2
(
τ
d
f
)
, where
τ
d
=
1
FSR
M
is
the delay on the MZI. Thus the periodic spectral minima of the PSD can be used to extract
the MZI FSR accurately. Fig. 2(a) gives the spectrum of the MZI outputs from 110 MHz - 200
MHz, which spans the 17th to 29th spectral minima of the MZI. The frequency location of each
minima was divided by its order to create the plot of MZI FSR values in Fig. 2(b). The average
MZI FSR is 6.723 MHz
±
2.7kHz. The offset frequency,
Δ
f
, is about 1 - 5 times the FSR of
the MZI, which means
T
d
Tm
1 - 5. As a result, the error contribution from the second term in
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−0.2
−0.1
0
0.1
0.2
0.3
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time (ms)
Transmission
a.u.)
w/o PM
w/ PM
MZI
Δ
f
PM freq: f
m
(f
m
<FSR)
FSR = f
m
+
Δ
f
(a)
(b)
Fig. 1.
Experimental setup.
(a) A schematic is shown for the sideband spectroscopy
method used to measure dispersion. A phase modulator (PM) creates sidebands on a probe
laser that are set to coincide approximately with the cavity FSR. Also, a Mach-Zehnder
interferometer creates a reference spectrum to measure the offset frequency
Δ
f. The laser
is scanned so as to produce the spectrum shown in panel (b). (b) Schematic traces of the
sideband spectroscopy are shown. When the phase modulation is “off” the red trace is ob-
served showing that the laser is scanning through a single cavity resonance. With the phase
modulation “on” and with its frequency set to be close in value to the cavity FSR, three
spectral peaks appear as the two phase modulation sidebands scan through their respective
cavity resonances (neighboring the resonance probed by the scan-laser, carrier wave). By
using the green interferometer trace to measure the offset
Δ
f and adding this offset to the
phase modulation frequency, the cavity FSR can be measured.
Eq. (1) is no larger than
±
15 kHz.
Finally, we consider the error contributions from the third and fourth terms in Eq. (1). In
practice, these errors are on the order of 100 kHz in our measurement, based on repetitive
measurements of the cavity FSR for one specific cavity mode using the method described above.
For instance, 10 measurement ofa6mmdisk resontor with a cavity Q of 200 million gives
an FSR of 10.8230 GHz
±
109kHz. As a result, the overall uncertainty in measurement of
microcavity FSR is set by these contributions.
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110
120
130
140
150
160
170
180
190
200
−100
−95
−90
−85
−80
−75
−70
−65
−60
Frequency (MHz)
Power (dBm)
16
18
20
22
24
26
28
30
6.717
6.718
6.719
6.72
6.721
6.722
6.723
6.724
6.725
6.726
6.727
Order
FSR of MZI (MHz)
(a)
(b)
17th
29th
17th
29th
Fig. 2.
Measurement of the FSR of the Mach Zehnder interferometer (MZI).
(a) Meas-
ured power spectrum of the photocurrent output from a balanced photodetector whose in-
puts detect the complementary outputs of the MZI. For this measurement the laser fre-
quency is close to a quadrature point of the MZI and the spectral measurement extends
from 110-200 MHz. (b) FSR of the MZI extracted from each order in (a). The dashed line
is the average.
3. Cavity transverse mode spectroscopy
We apply the above approach to measure the FSR of different transverse modes of the same cav-
ity. Comparison to finite element modeling then enables a mapping of spectral peaks with spa-
tial modes of the resonator. The TE modes of a 6 mm wedge resonator (FSR of approximately
10.8 GHz) are characterized. Details on the fabrication and properties of this resonator are pre-
sented in [10]. Briefly, however, these silica-based devices are fabricated on a silicon wafer
using only lithography and a combination of wet and dry etching. They have optical Q factors
ranging from several hundred million to nearly 1 billion. Their cross section is wedge-like and
the device characterized in Fig. 3 has a wedge angle
α
12
in a silica oxide with thickness
T
10
μ
m (measured using an SEM). Schematic cross sections of the device showing three
transverse modes are illustrated in Fig. 3(d). Optical coupling uses a fiber taper coupler [26,27].
In Fig. 3(a) a spectral scan encompassing one FSR is shown. By scanning over two FSRs
it is possible to identify repeating peaks that are associated with a particular transverse mode
family. The various spectral peaks shown correspond to distinct transverse modes of the device,
and are labeled from A to J. In Fig. 3(b) a “zoom-in” spectrum in the vicinity of peak G is
shown to illustrate the measurement technique detailed in Fig. 1. Also shown in Fig. 3(b) is the
Mach-Zehnder reference trace. Measurement results giving the FSR for each transverse mode
are provided in Fig. 3(c) (data are circles). Note that the transverse mode dispersion introduces
a difference of about 1 MHz in the FSR of neighboring transverse modes. This difference is
easily resolved as the measurement uncertainty is about 100 kHz (see discussion in previous
section). In order to identify the cavity modes, we calculate the FSR of different transverse-
order modes by FEM simulation in a commercial FEM solver (COMSOL multiphysics). The
simulation method is based on the 2D simulation of the whispering-gallery modes (WGM) of
axisymmetric resonators described in [28]. The calculated FSR values are plotted in Fig. 3(c)
for three, slightly-different, cavity dimensions (wedge angle and disk thickness). Several other
geometries were also simulated, however, the three shown gave the best agreement over the
range of modes measured. From the three FEM plots (indicated by triangles, diamonds, and
squares), it can be seen that a small perturbation of the cavity dimension maintains the FSR
sequence of different orders. Moreover, the three FEM plots are in reasonable agreement with
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1
2
3
4
5
6
7
8
9
10
11
0
2
4
6
8
10
12
14
16
18
20
Transverse mode order
FSR - 10.8230 GHz (MHz)
α
=12.2,T=9.7
μ
m
α
=12.0,T=9.5
μ
m
α
=12.0,T=9.7
μ
m
data
−4
−2
0
2
4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Frequency sweep (GHz)
Transmission (a.u.)
−10
−5
0
5
10
−0.04
−0.02
0
0.02
0.04
0.06
0.08
Relative frequency (MHz)
Transmission (a.u.)
G
C
J
D
I
A
F
E
H
B
A
B
C
D
E
F
H
I
J
G
(a)
(b)
(c)
(d)
1st
1st
5th
5th
9th
9th
10 m
m
10 m
μ
Fig. 3.
Cavity transverse mode spectroscopy
(a) Transmission spectrum fora6mm
wedge resonator. Multiple transverse modes (labeled from A to J) are shown within the
frequency sweep of one FSR (10.8 GHz). The lower green trace is the Mach-Zehnder ref-
erence inteferometer (MZI) (FSR of the MZI is 6.723 MHz, MZI fringes are resolved in
panel (b)). By using the MZI fringes and the calibrated MZI FSR, the original horizontal
axis (time span, as shown in Fig. 1(b)) can be converted to frequency span. (b) Zoom-in
measurement of the peak G in panel (a). The Lorentzian fit shows a loaded cavity linewidth
of 1.03 MHz. This is also the fundamental mode indicated in panels (c) and (d). (c) Mapping
of the cavity transverse-order to each spectral peak by comparing the FSR measurement
with FEM simulation. Three slightly different cavity geometries are used for FEM simula-
tion, and the FSR of the simulated transverse modes maintains the sequence regardless of
geometry. (d) Intensity profile of the 1st, 5th and 9th transverse-order modes calculated by
FEM. The corresponding spectral peaks are given to the right of the profile.
the data. (The 7th transverse mode order is not identified. The reason may be due to unfavor-
able coupling position or phase match of taper-fiber coupling). Through this comparison it was
possible to map each spectral peak to a particular transverse mode family. Several identifica-
tions have been made in Fig. 3(d), including the fundamental mode “G”. It is interesting to note
that this fundamental mode has the smallest FSR of all of the modes. This is expected since
the fundamental mode should also exhibit the largest effective index. Also, the slight plateau in
the FSR versus mode order (see Fig. 3(c)) is associated with the higher-order-modes extending
radially inward beyond the wedge region.
This mapping is based on the increment of the FSR of each transverse mode order; and if the
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1
1.2
1.4
1.6
1.8
2
−50
−40
−30
−20
−10
0
10
20
30
40
50
Wavelength ( m)
D (ps/nm/km)
α
= 10
= 20
= 30
°
°
°
α
α
Geometric
Total
Material
μ
(a)
(b)
(c)
(d)
10 m
m
10 m
μ
10
°
20
°
30
°
1
1.2
1.4
1.6
1.8
2
−60
−40
−20
0
20
40
60
80
Wavelength (
μ
m)
Δ
FSR (kHz)
= 10
= 20
= 30
°
°
°
α
α
α
−60
−40
−20
0
20
40
60
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Relative M number
FSR−32.9382GHz (MHz)
Fig. 4.
Measurement of dispersion at two wavelengths for three cavity geometries.
(a)
Measured cavity
FSR
(wrt 32.9382 GHz) for 2 mm resonator (
α
20
, oxide thickness,
T
8
μ
m) plotted versus relative azimuthal mode number M around the 1550 nm spectral
region. The dashed, red line is a linear fit giving 12.2 kHz/FSR dispersion. (b) Measured
(colored makers) and simulated cavity dispersion,
Δ
FSR
, as a function of wavelength for
2 mm disk resonators with three wedge angles (
α
10
,20
and 30
,T
8
μ
m). (c)
Solid lines give the dispersion parameter,
D
, converted from the
Δ
FSR
values in (b), using
Δ
FSR
c
2
λ
2
D
4
π
2
n
3
R
2
[29]. The dashed line is the silica material dispersion from the Sellmeier
equation. The three dotted lines are the geometric dispersion, obtained by subtracting the
material dispersion from the total dispersion. The measurement data points are given as
markers. (d) Intensity profile for 2 mm resonators with 10
,20
and 30
wedge angles.
FSRs between two neighboring pairs of transverse modes are close to each other, for example,
the 4-6th modes in Fig. 3(c), the modes may not be mapped accurately. The labeling of spectral
peaks “D, A, I” to mode order 4, 5, and 6 is therefore only one possible mapping. However,
having the FSR of each transverse mode is important by itself and can differentiate between
low order and high order modes. It can be particularly usefully in studying stimulated Brillouin
lasers [11] and microcomb generation in microresonators [8,22].
4. Dispersion characterization of wedge disk resonators
To measure the dispersion within a single transverse mode family, the above technique is re-
peated for a sequence of spectral peaks having the same transverse mode order. However, be-
cause the FSR dispersion of these resonators,
Δ
FSR
=
FSR
m
FSR
m
1
, is usually very small
(in the order of 1-10 kHz/FSR, depending on the resonator diameter), it is necessary to mea-
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sure the cavity FSR over multiple FSR separations. Fig. 4(a) shows the cavity FSR for a 2 mm
cavity (
α
20
,T
8
μ
m) plotted versus the relative cavity azimuthal mode number M (FSR
number) measured around the 1550 nm region. The dashed line is a linear fit which gives a
dispersion of 12.2 kHz/FSR. This value is in good agreement with the simulated value (12.8
kHz/FSR, simulation is based on the 2D FEM solver of the whispering-gallery modes of ax-
isymetric microcavities as described in [28]). The RMS error of the FSR measurement is 180
kHz (for a cavity Q of 150 million), which gives a relative precision of 5
.
5
×
10
6
for a cavity
FSR of 32.9382 GHz. In Fig. 4(b) measurements of the FSR dispersion of disk resonators with
three different wedge angles (
α
10
,20
,30
) and at two different wavelengths (1550nm and
1310nm) are presented along with the calculated
Δ
FSR
. Fig. 4(c) converts the FSR dispersion
values to the more widely used dispersion parameter,
D
d
d
λ
(
1
v
g
)
, using
Δ
FSR
c
2
λ
2
D
4
π
2
n
3
R
2
[29],
where R is the cavity radius, n is the refractive index and
λ
is the wavelength. Also, the mate-
rial dispersion parameter,
D
M
(calculated from the Sellmeier equation [30]), and the geometric
dispersion parameter (
D
G
=
D
D
M
) are plotted. From Fig. 4(b) and (c), it can be seen that the
measurement data (in markers) agree reasonably well with the simulation. Also, note that res-
onators having smaller wedge angles feature larger geometrical dispersion. This is consistent
with the centroid of the mode’s “orbit” around the resonator being shifted radially inward as
wavelength increases, and hence smaller wedge angles enhancing this tendency.
5. Conclusion
In conclusion, we have demonstrated a simple and accurate approach to measure the FSR of
microcavities by introducing external phase modulation and frequency calibration with an MZI.
This FSR measurement method is applicable for high-Q resonators with microwave rate FSR
and has a precision of 5
.
5
×
10
6
(given a cavity Q of 150 million). We have used this approach
to measure the FSRs of different cavity transverse modes and find that the cavity transverse
mode can be identified from their FSRs when comparing with FEM simulation results. Finally,
we have applied the FSR-measurement approach to characterize cavity dispersion, which is
important in many nonlinear photonic applications, such as the generation of microcombs.
Acknowledgments
The authors acknowledge the financial support from the DARPA QuASAR program, the Insti-
tute for Quantum Information and Matter, an NSF Physics Frontiers Center with support of the
Gordon and Betty Moore Foundation, NASA and the Kavli NanoScience Institute.
#176269 - $15.00 USD
Received 17 Sep 2012; revised 31 Oct 2012; accepted 1 Nov 2012; published 7 Nov 2012
(C)
2012
OSA
19
November
2012
/ Vol.
20,
No.
24
/ OPTICS
EXPRESS
26344