# Optical frequency comb generation from a monolithic microresonator

###### Abstract

Optical frequency combsUdem2002 (); Cundiff2003 (); BookFemto2005 () provide
equidistant frequency markers in the infrared, visible and
ultra-violetJones2005 (); Gohle2005 () and can link an unknown optical
frequency to a radio or microwave frequency referenceDiddams2000 (); Diddams2001 (). Since their inception frequency combs have triggered major
advances in optical frequency metrology and precision
measurementsDiddams2000 (); Diddams2001 () and in applications such as
broadband laser-based gas sensingThorpe2006 () and molecular
fingerprintingDiddams2007 (). Early work generated frequency combs by
intra-cavity phase modulationKOUROGI1993 (); Ye1997 (), while to date
frequency combs are generated utilizing the comb-like mode structure of
mode-locked lasers, whose repetition rate and carrier envelope phase can be
stabilizedJones2000a (). Here, we report an entirely novel approach in
which equally spaced frequency markers are generated from a continuous wave
(CW) pump laser of a known frequency interacting with the modes of a
monolithic high-Q microresonatorArmani2003 () via the Kerr
nonlinearityKippenberg2004a (); Savchenkov2004a (). The intrinsically
broadband nature of parametric gain enables the generation of discrete comb
modes over a 500 nm wide span ( THz) around 1550 nm without relying
on any external spectral broadening. Optical-heterodyne-based measurements
reveal that cascaded parametric interactions give rise to an optical frequency
comb, overcoming passive cavity dispersion. The uniformity of the mode spacing
has been verified to within a relative experimental precision of
. In contrast to femtosecond mode-locked
lasersSteinmeyer1999 () the present work represents an enabling step
towards a monolithic optical frequency comb generator allowing significant
reduction in size, cost and power consumption. Moreover, the present approach
can operate at previously unattainable repetition ratesKeller2003 ()
exceeding 100 GHz which are useful in applications where the access to
individual comb modes is required, such as optical waveform
synthesisWeiner2000 (), high capacity telecommunications or astrophysical
spectrometer calibrationmurphyar2007 ().

###### pacs:

PACS number: 42.65.Sf, 42.50.VkOptical microcavitiesVahala2003 () are owing to their long temporal and
small spatial light confinement ideally suited for nonlinear frequency
conversion, which has led to a dramatic improvement in the threshold of
nonlinear optical light conversionBookchang (). In contrast to stimulated
gain, parametric frequency conversionDunn1999 () does not involve coupling
to a dissipative reservoir, is broadband as it does not rely on atomic or
molecular resonances and constitutes a phase sensitive amplification process,
making it uniquely suited for tunable frequency conversion. In the case of a
material with inversion symmetry - such as silica - the non linear optical
effects are dominated by the third order non linearity. The process is based
on four-wave mixing among two pump photons (frequency ) with a signal
() and idler photon () and results in the emergence of
(phase coherent) signal and idler sidebands from the vacuum fluctuations at
the expense of the pump field (cf. Fig.1). The observation of parametric
interactions requires two conditions to be satisfied. First momentum
conservation has to be obeyed. This is intrinsically the case in a whispering
gallery type microcavityVahala2003 () since the optical modes are angular
momentum eigenstates and have (discrete) propagation constants resulting from the periodic boundary condition, where the
integer designates the mode number and denotes the cavity radius.
Hence the conversion of two pump photons (propagation constant )
into adjacent signal and idler modes (, , ) conserves momentum
intrinsicallyKippenberg2004a () (analogous reasoning applies in the case
where the two annihilated photons are in different modes, i.e. for four-wave
mixing, cf. Fig. 1b). The second condition that has to be met is energy
conservation. As the parametric process creates symmetrical sidebands with
respect to the pump frequency (obeying , where
is the Planck constant) it places stringent conditions on the cavity
dispersion that can be tolerated since it requires a triply resonant cavity.
This is a priori not expected to be satisfied, since the distance between
adjacent modes
(the free spectral range, FSR) can vary due to both material and intrinsic
cavity dispersion which impact and thereby render optical
modes (having frequencies , where is the speed of light in vacuo) non-equidistant.
Indeed, it has only recently been possible to observe these processes in
microcavities (made of crystallineSavchenkov2004a () CaF and
silicaKippenberg2004a (); Carmon2007 ()).

Importantly, this
mechanism could also be employed to generate optical frequency combs: the
initially generated signal and idler sidebands can interact among each other
and produce higher order sidebands (cf. Fig. 1) by non-degenerate four-wave
mixing (FWM)STOLEN1982 () which ensures that the frequency difference of
pump and first order sidebands is exactly
transferred to all higher order sidebands. This leads to an equidistant
spectrum throughout the entire comb. This can be readily seen by
noting that e.g. the 2 order sidebands are generated by mixing
among the pump and first order signal/idler sidebands (e.g. ), which rigidly determines the
spacing of any successively higher sidebands. Thus, provided the cavity
exhibits low dispersion, the successive four-wave mixing to higher orders
would intrinsically lead to the generation of phase coherent sidebands with
equal spacing, i.e. an optical frequency comb. Here, we report that
microresonators allow realization of this process and generation of optical
frequency combs.

We employ ultra-high-Q monolithic microresonators in the form of silica toroidal microcavitiesArmani2003 () on a silicon chip, which possess giant photon storage times () i.e. ultra-high quality factors () and small mode volumes. Highly efficient coupling is achieved using tapered optical fibersSpillane2003 (). Owing to the high circulating power, parametric interactions are readily observed at a threshold of approx. 50 W. When pumping with a continuous wave (CW) 1550-nm laser source, we observe a massive cascade and multiplication of the parametric sidebands extending to both higher and lower frequencies. Fig. 1a shows a spectrum in which a 75-m-diameter microcavity was pumped with 60 mW power, giving rise to an intra-cavity intensity exceeding 1 GW/cm. The parametric frequency conversion could extend over more than 490 nm (cf. Fig 1a inset), with the total conversion efficiency being 21.2 % (The highest observed conversion efficiency was 83 % by working in the over-coupled regimeArmani2003 ()). These bright sidebands (termed Kerr combs in the remaining discu ssions) could be observed in many different samples. Also, in the largest fabricated samples (190 m diameter) 380-nm broad Kerr combs comprising 134 modes spaced by 375 GHz could be generated at the expense of slightly higher pump power (cf. Supplementary Information).

To verify that the Kerr comb indeed contains equidistant
frequencies, we employed a fiber-laser based optical frequency
combKubina2005 () from Menlo Systems (termed “reference
comb” in the remaining discussion) as a reference grid whose
repetition rate is MHz. The principle underlying our
measurement is that the beating generated on a photodiode by superimposing the
reference comb with the Kerr comb will produce beat notes which constitute a
unique replica of the optical spectrum in the radio frequency domain, provided
that the highest produced beat note in the detection process is
(cf. Fig. 2b), similar to multi-heterodyne frequency
comb spectroscopySchliesser2005 (). Specifically, if the Kerr comb is
equidistant, the beat notes with the reference comb will constitute an
equidistant comb in the RF domain (with frequency spacing , where
). Fig.
2a shows the experimental setup for the optical beat measurement. In brief, an
external cavity laser at 1550 nm was used as pump laser (cf. Fig 2 main
panel). The Kerr lines of the microcavity were superimposed with the reference
comb in a beat note detection unit (BDU), consisting of polarizing optics to
combine the reference and Kerr combs and a grating to select the desired
region of spectral overlap. In this manner, the beating of 9 simultaneously
oscillating parametric modes (covering nm of wavelength span) were
recorded, as shown in Fig. 2c. Remarkably, from the equidistant spacing of the
radio-frequencies, it is found that the generated sidebands are
equidistant to within less than 5 kHz (as determined by the
measurement time of 200 s).

To improve the accuracy, we developed an additional experiment where we measured the beat-notes of three Kerr modes with the fiber-reference comb using three separate BDUs (cf. Fig. 3a) each counting a single beat of radio frequency (, , ). A signal-to-noise ratio exceeding 30 dB in 500 kHz bandwidth was achieved, sufficient to use radio-frequency counters, which were all referenced to a 10 MHz reference signal provided by the MPQ hydrogen Maser. The beat-note measured on BDU1 was used to implement an offset lock between a single reference comb line and the pumping laser by a known offset frequency (). The second (third) counter measured the N (M) mode of the Kerr comb as shown in Fig. 3. For equidistant mode spacing, the second (third) BDU gives rise to the beat frequency (). The uniformity of the Kerr comb was then checked by comparing the variation in the mode spacing, i.e. . Alternatively, direct counting of the ratio was implemented (using frequeny mixing and ratio counting, cf. SI). Fig. 4b shows the result of this measurement (for , ) and a counter gate time () of 1 second and more than 3000 records. The scatter in the data follows a Gaussian distribution (and follows a dependence of the Allan deviation, cf. Fig. 4b). The cavity modes of this measurement span over approx. 21 nm and yield a deviation from the mean of mHz. Note that the wavelength span that could be used for the measurement is currently limited by the gain bandwidth of an EDFA, which had to be used to amplify the reference comb to have sufficient power to run three BDUs simultaneously. The results for different gate times and for the two different counting methods are shown in Table 1 (the complete list is contained in the supplementary information). The weighted average of these results verifies the uniformity of the comb spacing to a level of (when referenced to the optical carrier). Normalized to the bandwidth of the measured Kerr lines (2.1 THz), this corresponds to . This accuracy is on par with measurements for fiber based frequency combsKubina2005 () and confirms that the generated Kerr combs exhibit uniform mode spacing.

N | Mean () | Technique | |||
---|---|---|---|---|---|

1 | 3493 | -0.9 5.5 | 322 | 2.710 | 2 Counters |

3 | 173 | 5.8 12.6 | 165 | 6.310 | Ratio |

10 | 22 | -17.9 15.0 | 70 | 7.510 | Ratio |

30 | 39 | 1.7 7.4 | 46 | 3.710 | Ratio |

100 | 42 | -0.3 2.7 | 17 | 1.410 | Ratio |

300 | 14 | -0.8 2.8 | 11 | 1.410 | Ratio |

Next we investigated the role of dispersion underlying the
observed comb generation. Dispersion in whispering-gallery-mode (WGM)
microcavities is characterized by the deviation in the free spectral range
and has two
contributions. Geometrical dispersion accounts for a negative FSR dispersion,
given by where the cavity radius (cf. supplementary information).
Material dispersion on the other hand is given by , where
is the group velocity dispersion parameter. Since the GVD of silica is
positive for wavelength greater than 1.3 m (anomalous dispersion), it can
compensate the intrinsic resonator dispersion (causing ). Indeed we measured a positive dispersion (cf. SI) which
equates to only ca. 20 MHz over a span of ca. 60 nm. This low value indicates
that the present experiments are carried out close to the zero dispersion
wavelength, in agreement with theoretical predictions.

Note that the
residual cavity dispersion exceeding the “cold” cavity linewidth does not preclude the parametric comb
generation process. This can be explained in terms of a nonlinear optical mode
pulling process as reported in Ref. Kippenberg2004a (). The strong CW pump
laser will induce both self-phase modulation (SPM) and cross-phase modulation
(XPM)agrawal (), the latter being twice as large as the former. The
resultant XPM and SPM induced refractive index changes will shift the cavity
resonance frequencies by different amounts, thereby causing a net change in
the (driven) cavity dispersion from its passive (un-driven)
valueKippenberg2004a (). This nonlinear mode pulling can provide a
mechanism to compensate the residual cavity dispersion.

Regarding
future experimental work in light of applications in metrology, we note that
absolute referencing can be attained by locking the pump laser to a known
atomic transition and locking the mode spacing to a microwave reference (such
as a Cs atomic clock). The latter requires that the two degrees of freedom of
the comb, its repetition rate (i.e. mode spacing, ) and frequency
offset, i.e. to be controlled independently. Indeed it could already been
shown in a proof of concept experiment that it is possible to lock two modes
of the Kerr comb simultaneously to two modes of the reference comb, showing
that we are able to control both and . The two
actuators used for this lock are the detuning of the pump laser from the
microcavity resonance and the pump power, which affects the optical pathlength
of the cavity via the thermal effect and the nonlinear phase shift.

Pertaining to the implications of our work, we note that the present
observation of a monolithic frequency comb generator could potentially prove
useful for frequency metrology, given however further improvements. Evidently
a readily measurable repetition rate would prove useful when directly
referencing the optical field to a microwave signalCundiff2003 (). To this
end a 660-m-diameter microcavity would already allow operating at
repetition rates GHz, which can be detected using fast photodiodes. On
the other hand, a large mode spacing as demonstrated here could prove useful
in several applications, such as line-by-line pulse shaping, calibration of
astrophysical spectrometers or direct comb spectroscopy. The high repetition
rate from an on chip device may also prove useful for the generation of
multiple channels for high capacity telecommunications (spacing 160 GHz) and
for the generation of low noise microwave signals. Furthermore, we note that
parametric interactions do also occur in other types of microcavities - e.g.
CaFSavchenkov2004a () - provided the material exhibits a third order
nonlinearity and sufficiently long photon lifetimes. As such the cavity
geometry is not conceptually central to the work and the reported phenomena
should become equally observable in other types of high-Q microresonators,
such as silicon, SOI or crystalline based WGM-resonators. Indeed the recent
observation of net parametric gainFoster2006 () on a silicon chip is a
promising step in this direction.

Acknowledgements: The authors thank T. W. Hänsch, Th. Udem, K. J. Vahala and Scott Diddams for critical discussions and suggestions. TJK acknowledges support via an Independent Max Planck Junior Research Group. This work was funded as part of a Marie Curie Excellence Grant (MEXT-CT-2006-042842), the DFG funded Nanoscience Initiative Munich (NIM) and a Marie Curie Reintegration Grant (MIRG-CT-2006-031141). The authors gratefully acknowledge J. Kotthaus for access to clean-room facilities for sample fabrication.

## References

- [1] T. Udem, R. Holzwarth, and T. W. Hansch. Optical frequency metrology. Nature, 416(6877):233–237, March 2002.
- [2] S. T. Cundiff and J. Ye. Colloquium: Femtosecond optical frequency combs. Reviews Of Modern Physics, 75(1):325–342, January 2003.
- [3] S. T. Ye, J. & Cundiff. Femtosecond Optical Frequency Comb: Principle, Operation and Applications. Springer, 2005.
- [4] R. J. Jones, K. D. Moll, M. J. Thorpe, and J. Ye. Phase-coherent frequency combs in the vacuum ultraviolet via high-harmonic generation inside a femtosecond enhancement cavity. Physical Review Letters, 94(19):193201, May 2005.
- [5] C. Gohle, T. Udem, M. Herrmann, J. Rauschenberger, R. Holzwarth, H. A. Schuessler, F. Krausz, and T. W. Hansch. A frequency comb in the extreme ultraviolet. Nature, 436(7048):234–237, July 2005.
- [6] S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hansch. Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb. Physical Review Letters, 84(22):5102–5105, May 2000.
- [7] S. A. Diddams, T. Udem, J. C. Bergquist, E. A. Curtis, R. E. Drullinger, L. Hollberg, W. M. Itano, W. D. Lee, C. W. Oates, K. R. Vogel, and D. J. Wineland. An optical clock based on a single trapped hg-199(+) ion. Science, 293(5531):825–828, August 2001.
- [8] M. J. Thorpe, K. D. Moll, R. J. Jones, B. Safdi, and J. Ye. Broadband cavity ringdown spectroscopy for sensitive and rapid molecular detection. Science, 311(5767):1595–1599, March 2006.
- [9] S. A. Diddams, L. Hollberg, and V. Mbele. Molecular fingerprinting with the resolved modes of a femtosecond laser frequency comb. Nature, 445(7128):627–630, February 2007.
- [10] M. Kourogi, K. Nakagawa, and M. Ohtsu. Wide-span optical frequency comb generator for accurate optical frequency difference measurement. Ieee Journal Of Quantum Electronics, 29(10):2693–2701, October 1993.
- [11] J. Ye, L. S. Ma, T. Day, and J. L. Hall. Highly selective terahertz optical frequency comb generator (vol 22, pg 301, 1997). Optics Letters, 22(10):746–746, May 1997.
- [12] D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff. Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis. Science, 288(5466):635–639, April 2000.
- [13] D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala. Ultra-high-Q toroid microcavity on a chip. Nature, 421(6926):925–928, February 2003.
- [14] T. J. Kippenberg, S. M. Spillane, and K. J. Vahala. Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity. Physical Review Letters, 93(8):083904, August 2004.
- [15] A. A. Savchenkov, A. B. Matsko, D. Strekalov, M. Mohageg, V. S. Ilchenko, and L. Maleki. Low threshold optical oscillations in a whispering gallery mode CaF2 resonator. Physical Review Letters, 93(24):243905, December 2004.
- [16] G. Steinmeyer, D. H. Sutter, L. Gallmann, N. Matuschek, and U. Keller. Frontiers in ultrashort pulse generation: Pushing the limits in linear and nonlinear optics. Science, 286(5444):1507–1512, November 1999.
- [17] U. Keller. Recent developments in compact ultrafast lasers. Nature, 424(6950):831–838, August 2003.
- [18] A. M. Weiner. Femtosecond pulse shaping using spatial light modulators. Review Of Scientific Instruments, 71(5):1929–1960, May 2000.
- [19] M. T. et al. Murphy. High-precision wavelength calibration with laser frequency combs. arXiv:astro-ph/0703622, 2007.
- [20] K. J. Vahala. Optical microcavities. Nature, 424(6950):839–846, August 2003.
- [21] A. J. Chang, R. K. & Campillo. Optical processes in microcavities. World Scientific, 1996.
- [22] M. H. Dunn and M. Ebrahimzadeh. Parametric generation of tunable light from continuous-wave to femtosecond pulses. Science, 286(5444):1513–1517, November 1999.
- [23] T. Carmon and K. J. Vahala. Visible continuous emission from a silica microphotonic device by third-harmonic generation. Nature Physics, 3(6):430–435, June 2007.
- [24] R. H. Stolen and J. E. Bjorkholm. Parametric amplification and frequency-conversion in optical fibers. Ieee Journal Of Quantum Electronics, 18(7):1062–1072, 1982.
- [25] S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J. Vahala. Ideality in a fiber-taper-coupled microresonator system for application to cavity quantum electrodynamics. Physical Review Letters, 91(4):043902, July 2003.
- [26] P. Kubina, P. Adel, F. Adler, G. Grosche, T. W. Hansch, R. Holzwarth, A. Leitenstorfer, B. Lipphardt, and H. Schnatz. Long term comparison of two fiber based frequency comb systems. Optics Express, 13(3):904–909, February 2005.
- [27] A. Schliesser, M. Brehm, F. Keilmann, and D. W. van der Weide. Frequency-comb infrared spectrometer for rapid, remote chemical sensing. Optics Express, 13(22):9029–9038, October 2005.
- [28] G.P. Agrawal. Nonlinear Fiber Optics. Academic Press, 2006.
- [29] M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta. Broad-band optical parametric gain on a silicon photonic chip. Nature, 441(7096):960–963, June 2006.
- [30] T. J. Kippenberg, S. M. Spillane, and K. J. Vahala. Demonstration of ultra-high-Q small mode volume toroid microcavities on a chip. Applied Physics Letters, 85(25):6113–6115, December 2004.
- [31] T. Carmon, L. Yang, and K. J. Vahala. Dynamical thermal behavior and thermal self-stability of microcavities. Optics Express, 12(20):4742–4750, October 2004.
- [32] T. J. Kippenberg, S. M. Spillane, and K. J. Vahala. Modal coupling in traveling-wave resonators. Optics Letters, 27(19):1669–1671, October 2002.
- [33] S. Schiller. Asymptotic-expansion of morphological resonance frequencies in Mie scattering. Applied Optics, 32(12):2181–2185, April 1993.
- [34] Tobias Kippenberg. Nonlinear Optics in Ultra-high-Q Whispering-Gallery Optical Microcavities. PhD thesis, California Institute of Technology, 2004.

## Appendix A Generation of Kerr Combs at lower repetition rates

Figure 5 shows the Kerr comb spectrum at a lower repetition rate mentioned in the main paper. The repetition rate is , corresponding to a free spectral range of . With larger samples it should be possible to generate repetition rates smaller than which permits the direct measurement of the repetition rate with high-bandwidth photodiodes.

## Appendix B Beat note experiments between the Fiber Laser Comb and Kerr Comb

To demonstrate the equidistant nature of the parametric Kerr lines,
a reference frequency comb in the form of a mode locked erbium fiber
laser is used (from Menlo Systems GmbH). The principle underlying
the measurement is similar to the concept of multi-heterodyne
spectroscopy[27]. Assuming that the
reference comb produces a spectrum with frequencies (where is the repetition rate,
is the carrier envelope offset frequency and is an integer number of order ) and the Kerr comb produces frequencies
( integer), the signal generated by
interfering the two combs will have an imprinted radio frequency
(RF) beat note spectrum. If the reference comb’s repetition rate is
adjusted such that a multiple of it is close to the Kerr mode
spacing, i.e. (with an integer ), then the different Kerr comb lines will generate
different RF beat notes which will again be evenly spaced, i.e.
their frequencies are (with and integer).

The experimental setup is depicted in Figure 2 of the main paper. A
tunable external cavity diode laser (ECDL) is used to pump a
microtoroid resonance as detailed in [30] and
[14]. Since the cavity resonances are polarization
dependent, a in-fiber polarization controller is used to adjust the
polarization of the pump laser. The microtoroid is placed in a
sealed enclosure containing a nitrogen atmosphere, to avoid the
deposition of water on the surface of the silica toroid which has
strong absorption bands in the regime. In the
microresonator a spectrum of modes is generated via nonlinear
parametric interactions and four-wave mixing (see main paper). The
output signal of the tapered optical fiber (containing the
parametric modes that are outcoupled from the microresonator back to
the tapered fiber) is split by two couplers and
monitored with a photodiode connected to an oscilloscope and an
optical spectrum analyzer. Another fraction of the taper output is
sent to a “beat detection unit” (BDU) and superimposed with a fiber-laser-based reference frequency
comb with a repetition rate of [26].
The BDU consists of quarter wave plates and half wave plates to
prepare orthogonal linear polarization in the two input beams, which
are subsequently combined using a polarizing beam splitter. By means
of a half-wave plate, an adjustable linear combination of the two
input beams’ polarizations is then rotated onto the transmission
axis of a polarizer, where the two input beams interfere. To
increase the signal-to-noise ratio (SNR), the spectral region
containing the Kerr comb lines is selected by a grating and finally
detected with a PIN InGaAs photodiode (Menlo Systems FPD 510). An
oscilloscope with a built-in FFT routine is utilized to analyze the
radio frequency spectrum. For rough analysis an electronic spectrum
analyzer is used. Since the repetition rate of the reference comb is
around the beat note frequencies between a laser
line and the reference comb are in the range of to
. Now the repetition rate of the reference comb is
adjusted until is a
small frequency such that for all of interest the condition is fulfilled. The
observation of an equidistant RF beat “comb” then provides proof for the equidistance of
the Kerr comb.

## Appendix C Measuring the accuracy of the mode spacing using heterodyne spectroscopy

### c.1 Measuring with two counters

To verify the equidistance of the Kerr comb modes it is necessary to know the frequencies of three Kerr comb modes simultaneously. The frequency counting is achieved by using radio frequency counters that are connected to a photodiode in a beat note detection unit (cf. figure 2 in the main paper). To determine the frequencies of three Kerr comb modes at the same time, three beat note detection units (BDU) have been built. By tuning the grating of the BDUs it is possible to measure the beat note frequency of a single Kerr comb line with a reference comb line. For simplicity reasons, one BDU is used to lock the diode laser pumping the microcavity to a single mode of the reference comb. Additionally the repetition rate of the reference comb is locked to a frequency of around 100 MHz, stabilized by a 10 MHz frequency standard generated by an in-house hydrogen maser. The two remaining beat detection units are placed at the output of the microcavity and the gratings are adjusted in a way that each of them counts a different Kerr comb mode. Note that the output of the reference comb had to be amplified with an EDFA to obtain sufficient power to run three BDUs simultaneously (a single line of the reference comb contains ca. 10 nW optical power). With this setup it was possible to achieve signal-to-noise ratios for the Kerr sideband beat notes of more than 30 dB at a resolution bandwidth of 500 kHz (Additional RF filters with a 3-dB-bandwidth of 3 MHz have been used to filter out background noise). In the present experiment we focused on counting the 5 (beat note frequency ) and the 7 Kerr comb sideband (beat note frequency ), whereas the pump laser was phase locked to the fiber laser reference comb such that its beat with the reference comb was fixed to a frequency . Note that the pump laser already constitutes one tooth of the Kerr comb. For an equally spaced Kerr comb we therefore expect and with and to be the beat note frequencies of the sidebands. The variation of the mode spacing of the Kerr comb is given by

(1) |

which is zero for an equally spaced comb. With the measured values for and and the known frequency it is possible to calculate the variation of the mode spacing . The two counters are referenced to the same frequency standard as the offset lock for the pump laser and are externally triggered with a signal from a pulse generator. This external triggering was necessary since the mode spacing of the Kerr comb was fluctuating by approximately 40 kHz r.m.s., giving rise to “breathing” of the Kerr comb modes(cf. figure 6). Hence, it proved cricital for a high accuracy that the two counters measured simultaneously, to allow the cancellation of the common fluctuations.

### c.2 Measuring the ratio of the distance to the sidebands

To avoid the synchronization problems mentioned before, the experimental setup depicted in figure 7 was used. In brief, the three counter signals were first electronically mixed with and filtered yielding only the distance between pump and the N (M) Kerr-sidebands. With this setup, a slightly smaller standard deviation of the measurements could be achieved by using just one counter with two inputs to measure the ratio of the distance between the pump beat and the two sideband beats,

(2) |

Solving this for and using equation 1 we obtain the dependence of the variation of the mode spacing from the ratio R:

(3) |

Using the frequency difference , which was set to approximately 10 MHz, it is possible to derive the variation of the mode spacing by measuring the frequency ratio R.

## Appendix D Experimental Results of the Counter Measurements

Table 2 shows the experimental results from the measurements of the Kerr comb equidistance. Note that a total of 9 data points out of the 8382 measurements from table 2 have been removed from analysis. These data points have been separated by the other data points of the respective measurements by more than 15 standard deviations. Assuming a Gaussian distribution (which was indeed found for the remaining 8373 measurements) the probability of measuring a point 15 standard deviations off as given by the cumulative error function is . These points are believed to originate from some local perturbations in the lab leading to a temporary reduction of the signal-to-noise level of the radio frequency beat notes. The weighted mean in table 2 has been calculated with the inverse squared standard error of the mean as weight:

(4) |

(5) |

The weighted mean calculated from all measurements leads to a variation of the modespacing of . Normalized to the optical carrier frequency of THz, this leads to an accuracy of the equidistance of .

Gate time (s) | Readings | Mean Value for (mHz) | StdDev of (Hz) | Counting Method |

0.03 | 217 | -33 556 | 8.2 | ratio |

0.1 | 223 | -80 181 | 2.7 | ratio |

0.3 | 293 | 2.4 50.1 | 0.86 | ratio |

1 | 3493 | -0.91 5.46 | 0.32 | 2 counters |

1 | 3499 | 3.9 10.1 | 0.60 | 2 counters |

1 | 98 | -40.1 27.4 | 0.27 | ratio |

1 | 179 | 8.0 25.5 | 0.34 | ratio |

3 | 173 | 5.8 12.6 | 0.17 | ratio |

10 | 22 | -17.9 15.0 | 0.070 | ratio |

30 | 39 | 1.65 7.41 | 0.046 | ratio |

60 | 72 | -1.88 3.00 | 0.025 | ratio |

100 | 18 | 1.12 5.98 | 0.024 | ratio |

100 | 42 | -0.26 2.69 | 0.017 | ratio |

300 | 14 | -0.82 2.83 | 0.011 | ratio |

Weighted Mean : | -0.8 mHz 1.4 mHz | - | - |

## Appendix E Measurement of Cavity Dispersion

To measure cavity dispersion, we employ the arrangement shown in figure 8. In brief, we first lock an external cavity laser around to one of the fundamental WGM cavity modes (the same resonance that gives rise to cascaded sidebands at higher power). The cavity resonance of the monolithic microresonator is locked to the external cavity laser by virtue of the thermal self locking technique[31]. The power is chosen to be far below the parametric threshold but sufficient to entail a stable lock. Next, the frequency comb is offset-locked to the external cavity laser by recording the beat note signal in a separate beat note detection unit (for working principle of the beat detection unit see last section). To achieve stable locking the generated beating is filtered and amplified yielding a SNR of ca. (at a resolution bandwidth of ). For dispersion measurement the frequency comb must be locked at an arbitrary detuning with respect to the ECDL. The latter is accomplished by mixing the beat note with a (variable) reference signal () down to and implementing a phase lock with feedback on the fiber comb’s repetition rate () by controlling the cavity length using a mirror mounted on a piezoelectric tube (Note that all RF generators and analyzers are stabilized using an in-house 10-MHz-reference). Owing to the fact that the cavity linewidth is and the repetition rate of the fiber comb (FC) is , not more than FC comb mode at a time can be resonant with one microresonator mode. Since measuring the coupling of an individual comb mode into the resonator in transmission is difficult, we measure the reflection of the cavity induced by modal coupling[32]. By variation of (and by recording simultaneously ) this allows to resolve the linewidth of individual cavity modes in reflection when using an OSA in zero-span mode. Hence this measurement provides an accurate means to measure frequency gap (free spectral range) between two cavity resonances and modulo the repetition rate of the fiber comb . The low power of the individual FC lines (ca. ) ensures that the probed cavity mode is not thermally distorted. To remove the ambiguity in the number of comb lines between the FSR of the cavity i.e. a second measurement was carried out with a different repetition rate, which allowed to retrieve . So the actual free spectral range between two cavity resonances can be derived by:

Figure 9 shows the experimental result of the dispersion measurement. The used cavity had a free spectral range (FSR) of nm, which corresponds to THz. Plotted in figure 9 is the accumulated dispersion of the FSR, which we express for convenience as . Here, the are the resonance frequencies of a “cold” microcavity. For this measurement, is a resonance at 1585 nm ( THz). From the graph it can be derived that the accumulated dispersion is 2.6 MHz per FSR (i.e. positive dispersion).

## Appendix F Dispersion Predictions

The dispersion in our microcavities has two contributions. First, whispering-gallery mode microcavities exhibit an intrinsic variation of the free spectral range owing to the resonator geometry. The resonance frequency of the fundamental mode of a microsphere is approximately given by [33]

(6) |

where is vacuum light speed, the refractive index, the cavity radius and the first zero of the Airy function (). Hence, the variation of the free spectral range

(7) |

is given by

(8) |

Evidently, the free spectral range reduces with increasing
frequency corresponding to a negative group velocity dispersion
(GVD), i. e. low frequency modes exhibit a shorter round trip time
than high frequency modes. Supplementary figure S3 shows the
variation for a 40- and 80-micron-radius microsphere.

A second contribution comes from the dispersion of the fused silica
material constituting the resonator. Its contribution can be
estimated by considering that the refractive index is actually a
function of frequency (and therefore mode number ), . Neglecting geometric dispersion, the GVD of fused silica
alone would lead to a FSR variation of

(9) |

where

(10) |

is the group-velocity dispersion of fused silica. This material parameter is well-known to change its sign in the 1300-nm wavelength region from about at 800 nm to at 1550 nm. Combining the two contributions, the positive sign of the GVD allows us in particular to cancel the geometric dispersion of our resonators to some extent, rendering the FSR nearly constant over a wide frequency span. Figure 10 displays the FSR variation for an 80- and 160-micrometer diameter microsphere, considering both material and geometric dispersion. Importantly, a zero dispersion point close to our operating wavelength occurs. Note that for a toroidal microcavity the location of the zero dispersion point is expected to be shifted to shorter wavelengths owing to the different resonator geometry. This expectation is borne out of finite element simulations showing that the resonance wavelength for a given value is shorter in a microtoroid cavity as compared to a microsphere [34].

## Appendix A: Symbols Used Throughout This Work

Symbols | Designation |
---|---|

Optical microcavity mode (with angular mode number) | |

Optical microcavity free spectral range | |

Optical microcavity variation of the free spectral range | |

Kerr comb carrier envelope offset frequency | |

Kerr comb mode spacing | |

Fiber reference comb repetition rate | |

Fiber reference comb carrier envelope frequency | |

Beat note unit (BDU) frequencies | |

Frequency spacing of the multi-heterodyne beat comb |