Pump frequency noise coupling into a
microcavity by thermo-optic locking
Jiang Li,
1
Scott Diddams,
2
and Kerry J. Vahala
1
∗
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology,
Pasadena, CA 91125, USA
2
Time and Frequency Division, National Institute of Standards and Technology,
Boulder, CO 80305, USA
∗
vahala@caltech.edu
Abstract:
As thermo-optic locking is widely used to establish a stable
frequency detuning between an external laser and a high Q microcavity, it
is important to understand how this method affects microcavity temperature
and frequency fluctuations. A theoretical analysis of the laser-microcavity
frequency fluctuations is presented and used to find the spectral dependence
of the suppression of laser-microcavity, relative frequency noise caused by
thermo-optic locking. The response function is that of a high-pass filter with
a bandwidth and low-frequency suppression that increase with input power.
The results are verified using an external-cavity diode laser and a silica disk
resonator. The locking of relative frequency fluctuations causes temperature
fluctuations within the microcavity that transfer pump frequency noise onto
the microcavity modes over the thermal locking bandwidth. This transfer
is verified experimentally. These results are important to investigations
of noise properties in many nonlinear microcavity experiments in which
low-frequency, optical-pump frequency noise must be considered.
© 2014 Optical Society of America
OCIS codes:
(140.4780) Optical resonators; (140.6810) Thermal effects; (140.3945) Micro-
cavities; (143.3325) Laser coupling.
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. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko,“Ultimate Q of optical microsphere resonators,” Opt.
Lett.
21
, 453–455 (1996).
4. 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).
5. 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. Photonics
6
, 369–373 (2012).
6. V. S. Ilchenko, and M. L. Gorodetskii, “Thermal nonlinear effects in optical whispering gallery microresonators,”
Laser Phys.
2
, 1004 (1992).
7. A. E. Fomin, M. L. Gorodetsky, I. S. Grudinin, and V. S. Ilchenko, “Nonstationary nonlinear effects in optical
microspheres,” J. Opt. Soc. Am. B
22
, 459–465 (2005).
8. T. Carmon, L. Yang, and K. J. Vahala, ”Dynamical thermal behavior and thermal self-stability of microcavities,”
Opt. Express
12
, 4742–4750 (2004).
9. T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced
Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett.
95
, 033901, (2005).
10. T. J. Kippenberg, and K. J. Vahala, “Cavity Optomechanics: Back-Action at the Mesoscale,” Science
321
, 1172–
1176 (2008).
#209930 - $15.00 USD
Received 10 Apr 2014; revised 30 May 2014; accepted 30 May 2014; published 5 Jun 2014
(C)
2014
OSA
16
June
2014
| Vol.
22,
No.
12
| DOI:10.1364/OE.22.014559
| OPTICS
EXPRESS
14559
11. M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, “A picogram-and nanometre-scale photonic-
crystal optomechanical cavity,” Nature
459
, 550–555 (2009).
12. Q. Lin, J. Rosenberg, X. Jiang, K. J. Vahala, and O. Painter, “Mechanical oscillation and cooling actuated by the
optical gradient force,” Phys. Rev. Lett.
103
, 103601 (2009).
13. P. DelHaye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and T. Kippenberg, “Optical frequency comb
generation from a monolithic microresonator,” Nature
450
, 1214–1217 (2007).
14. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-Based Optical Frequency Combs,” Science
322
, 555–559 (2011).
15. J. Li, H. Lee, T. Chen, and K. J. Vahala, “Low-Pump-Power, Low-Phase-Noise, and Microwave to Millimeter-
Wave Repetition Rate Operation in Microcombs,” Phys. Rev. Lett.
109
, 233901 (2012).
16. S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused-quartzmicroresonator optical
frequency comb,” Phys. Rev. A
84
, 053833 (2011).
17. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal
solitons in optical microresonators,” Nat. Photonics
8
, 145–152 (2014).
18. 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).
19. J. Li, H. Lee, and K. J. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun.
4
,
2097 (2013).
20. R. Drever, J. L. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, and H. Ward, “Laser phase and frequency
stabilization using an optical resonator,” Appl. Phys. B
31
, 97–105 (1983).
1. Introduction
High-Q microresonators have many applications that result from their combined low optical
loss and small mode volume [1–5]. These same properties also mean that thermal effects are
readily observable in microcavities. Thermal nonlinear effects in microresonators have been
studied previously, including thermal oscillation instability [6, 7], thermal linewidth broaden-
ing [6, 8], and thermal self-stability of cavity resonances [8]. Thermal self-stability of cavity
resonances, also called thermo-optic locking, maintains a stable detuning between an external
laser and a microcavity resonance [8]. The sign of stable detuning depends upon the sign of
d
L
OPL
/d
T
where
L
OPL
is the round-trip optical path length, and in silica a blue-detuned optical
signal is stable.
Thermo-optic locking (TOL) makes possible stable optical coupling to a narrow cavity reso-
nance without using any electronic feedback to control the laser frequency [8]. It is widely used
in microcavity experiments, including cavity optomechanics [9–12], microresonator-based fre-
quency combs [13–17], microcavity stimulated Brillouin lasers [5,18] and Brillouin microwave
synthesis [19]. However, TOL forces the resonant frequency of the microcavity to track any
fluctuations in the laser frequency that fall within the thermo-locking bandwidth. It is therefore
important to understand this mechanism as it can impact low-frequency fluctuations in these
systems.
In this paper, we study the relative frequency noise reduction between an external laser and a
microcavity under thermo-optic locking conditions. Using a small signal analysis, we derive the
laser-microcavity frequency-detuning fluctuation spectrum and then compare the predictions
to measurements of this spectrum using an external cavity diode laser and a 2 mm silica disk
resonator. The transfer of pump noise into other resonator modes is also verified experimentally.
2. Model of of laser-microcavity frequency noise
The coupled equations describing the cavity-mode amplitude,
a
, and cavity temperature change,
Δ
T
, can be written as [11],
̇
a
=[
i
Δ
−
ig
th
Δ
T
−
κ
2
]
a
+
√
κ
ex
s
(1)
Δ
̇
T
=
−
γ
th
Δ
T
+
Γ
|
a
|
2
(2)
#209930 - $15.00 USD
Received 10 Apr 2014; revised 30 May 2014; accepted 30 May 2014; published 5 Jun 2014
(C)
2014
OSA
16
June
2014
| Vol.
22,
No.
12
| DOI:10.1364/OE.22.014559
| OPTICS
EXPRESS
14560
where
a
is normalized such that
|
a
|
2
is intracavity field energy,
|
s
|
2
denotes the input signal
power,
Δ
≡
ω
l
−
ω
o
is the laser frequency (
ω
l
) detuning relative to the cold cavity resonance
(
ω
o
),
κ
(
κ
ex
) is the loaded (external) cavity optical energy decay rate.
g
th
=
−
dn
dT
ω
o
n
is the
thermo-optic tuning coefficient resulting from thermal changes in the refractive index (
n
), and
γ
th
is the temperature decay rate.
Γ
=
Γ
abs
/
c
th
is the heating rate due to optical absorption,
where
Γ
abs
is the component of the optical energy dissipation due to material absorption, and
c
th
is the heat capacity of the volume containing the cavity mode. Equation (2) can be understood
as an energy conservation relation: the net heat flow into the cavity (resulting in temperature
change) equals the heat flow into the cavity due to optical heating minus the heat flow out of
the cavity due to thermal conduction.
Small signal quantities are introduced as follows:
a
=
a
o
+
a
1
(
t
)
(3)
Δ
T
=
Δ
T
o
+
Δ
T
1
(
t
)
(4)
Δ
=
Δ
0
+
Δ
1
(
t
)
(5)
where in general
Δ
1
(
t
)
includes all sources of frequency fluctuation that change the differ-
ence of the laser frequency and cavity line center frequency, e.g. laser freqeuency noise, cavity
thermorefractive noise and photothermal noise. In this work, the dominant source of these fluc-
tuations is the laser frequency and the intensity noise contributions to the cavity are ignored in
the analysis. As an aside, the condition that frequencies of interest (within thermo lock band-
width) are much slower than the cavity bandwidth also allows introduction of a time-dependent
detuning variable in Eq. (5). These give the solutions for steady-state operation:
a
o
=
√
κ
ex
s
/
(
−
i
[
Δ
o
−
g
th
Δ
T
o
]+
κ
2
)
(6)
Δ
T
o
=
Γ
|
a
o
|
2
γ
th
=
Γ
γ
th
κ
ex
|
s
|
2
[
Δ
o
−
g
th
Δ
T
o
]
2
+
κ
2
/
4
(7)
and, upon linearization, the following time-dependent coupled system:
̇
a
1
=
[
i
Δ
o
−
ig
th
Δ
T
o
−
κ
2
]
a
1
+
i
[
Δ
1
−
g
th
Δ
T
1
]
a
o
(8)
̇
Δ
T
1
=
−
γ
th
Δ
T
1
+
Γ
(
a
o
a
∗
1
+
a
∗
o
a
1
)
(9)
Solving Eq. (7) gives the typical thermal bistability tuning curve of a microcavity [8]. Because
the cavity field damping rate,
κ
/
2, is typically much faster than the thermal damping rate of
the cavity,
γ
th
, the steady-state form of Eq. (8) can be used throughout the analysis (i.e., field
adiabatically follows the mode temperature and laser frequency fluctuations at rates less than
the cavity bandwidth):
a
1
=
i
[
Δ
1
−
g
th
Δ
T
1
]
a
o
−
i
[
Δ
o
−
g
th
Δ
T
o
]+
κ
2
(10)
Upon substitution of Eq. (10) into Eq. (9), the following equation results for the temperature
fluctuations of the optical mode driven by fluctuations of the laser detuning,
Δ
1
:
Δ
̇
T
1
=
−
[
γ
th
−
2
Γ
g
th
|
a
o
|
2
(
Δ
o
−
g
th
Δ
T
o
)
(
Δ
o
−
g
th
Δ
T
o
)
2
+
κ
2
4
]
Δ
T
1
−
2
Γ
|
a
o
|
2
(
Δ
o
−
g
th
Δ
T
o
)
(
Δ
o
−
g
th
Δ
T
o
)
2
+
κ
2
4
Δ
1
(11)
Solving this equation in the frequency domain gives,
Δ
̃
T
1
(
Ω
)=
−
2
Γ
|
a
o
|
2
(
Δ
o
−
g
th
Δ
T
o
)
(
Δ
o
−
g
th
Δ
T
o
)
2
+
κ
2
4
γ
th
−
2
Γ
g
th
|
a
o
|
2
(
Δ
o
−
g
th
Δ
T
o
)
(
Δ
o
−
g
th
Δ
T
o
)
2
+
κ
2
4
+
i
Ω
̃
Δ
1
(
Ω
)
(12)
#209930 - $15.00 USD
Received 10 Apr 2014; revised 30 May 2014; accepted 30 May 2014; published 5 Jun 2014
(C)
2014
OSA
16
June
2014
| Vol.
22,
No.
12
| DOI:10.1364/OE.22.014559
| OPTICS
EXPRESS
14561
10
0
10
1
10
2
10
3
10
4
10
5
10
6
−60
−50
−40
−30
−20
−10
0
10
Frequency
(Hz)
R(
Ω
) (dB)
K(
Ω
) (dB)
−60
−50
−40
−30
−20
−10
0
10
Pin 0.1 mW
Pin 0.2 mW
Pin 0.5 mW
Pin 1mW
Fig. 1. Plot of thermo-optic lock, laser-microcavity, relative-frequency-noise transfer func-
tion (
R
(
Ω
)
) and the cavity-frequency-noise transfer function (
K
(
Ω
)
) for several input laser
power levels at a warm-cavity detuning of
Δ
o
−
g
th
Δ
T
o
=
κ
/
2. Parameters used in these
plots are
γ
th
/
2
π
= 100 Hz, and
α
/
2
π
= 10 kHz/mW.
On the other hand, the laser-microcavity relative frequency fluctuation in the frequency domain
is
̃
Δ
lc
1
(
Ω
)=
−
g
th
Δ
̃
T
1
(
Ω
)+
̃
Δ
1
(
Ω
)
(13)
Substituting Eq. (12) into Eq. (13), the spectral density of the laser-microcavity relative fre-
quency fluctuation,
S
lc
Δ
ω
(
Ω
)
≡|
̃
Δ
lc
1
(
Ω
)
|
2
, is given by,
S
lc
Δ
ω
(
Ω
)=
γ
2
th
+
Ω
2
[
γ
th
+
α
P
in
]
2
+
Ω
2
S
Δ
ω
(
Ω
)
(14)
where
α
=
dn
dT
ω
o
n
Γ
2
κ
ex
(
Δ
o
−
g
th
Δ
T
o
)
[
(
Δ
o
−
g
th
Δ
T
o
)
2
+
κ
2
4
]
2
(15)
and
S
Δ
ω
(
Ω
)
≡|
̃
Δ
1
(
Ω
)
|
2
is the spectral density of the free-running laser frequency fluctuations.
Note that
Δ
o
−
g
th
Δ
T
o
is the steady-state detuning for the warm cavity. We can define the TOL
laser-microcavity relative frequency noise transfer function as:
R
(
Ω
)=
γ
2
th
+
Ω
2
[
γ
th
+
α
P
in
]
2
+
Ω
2
(16)
This expression has the form of a high-pass filter response governed by a zero at
γ
th
and a
power-dependent pole at
γ
th
+
α
P
in
. The transfer function is plotted in Fig. 1 for a range of
input laser power levels at a constant detuning. On account of the power-dependent pole, there
can be a substantial suppression of the noise. It is worth commenting on the impact of cavity
Q (or photon damping rate
κ
) on the TOL bandwidth and frequency-noise suppression. As
κ
ex
and
Δ
o
−
g
th
Δ
T
o
are both typically on the order of
κ
, it follows that
α
∼
1
/
κ
2
. Therefore, a
high optical Q greatly enhances the TOL bandwidth and frequency noise suppression factor.
#209930 - $15.00 USD
Received 10 Apr 2014; revised 30 May 2014; accepted 30 May 2014; published 5 Jun 2014
(C)
2014
OSA
16
June
2014
| Vol.
22,
No.
12
| DOI:10.1364/OE.22.014559
| OPTICS
EXPRESS
14562
Physically,
R
(
Ω
)
at
Ω
=
0 captures the so-called thermal-locking behavior of a high-Q micro-
cavity on account of the temperature dependence of refractive index. In addition, however, the
frequency response contained in
R
(
Ω
)
shows how this locking behavior is bandwidth limited
to a range determined by the thermal damping rate.
While the laser-microcavity, relative frequency fluctuations are suppressed due to thermo-
optic lock, the absolute frequency of cavity and the mode temperature of the cavity experience
fluctuations from the laser as a result of the locking. The microcavity temperature fluctuation
spectrum can be derived from Eq. (12):
S
Δ
T
(
Ω
)=
[
α
P
in
]
2
/
g
2
th
[
γ
th
+
α
P
in
]
2
+
Ω
2
S
Δ
ω
(
Ω
)
(17)
Accordingly, the cavity resonance frequency fluctuation spectrum is given by:
S
cav
Δ
ω
(
Ω
)=
g
2
th
S
Δ
T
(
Ω
)=
[
α
P
in
]
2
[
γ
th
+
α
P
in
]
2
+
Ω
2
S
Δ
ω
(
Ω
)
(18)
We define the TOL cavity frequency noise transfer function as:
K
(
Ω
)=
[
α
P
in
]
2
[
γ
th
+
α
P
in
]
2
+
Ω
2
(19)
K
(
Ω
)
is a low pass filter function (a power dependent pole at
γ
th
+
α
P
in
) that gives the transfer
of laser frequency noise into cavity frequency noise. The transfer function is also plotted in
Fig. 1 (right vertical axis) for a series of input laser power levels at constant detuning. It can
be seen that an increase of the input laser power not only gives a higher suppression of laser-
microcavity relative frequency fluctuations, it also causes the microcavity resonance to track
the external input laser more tightly with a larger bandwidth.
3. Measurement of laser-microcavity relative-frequency-noise suppression
The laser-microcavity relative-frequency-noise is contained in the power fluctuations transmit-
ted past the cavity. Specifically, the cavity resonance lineshape acts as a frequency discriminator
to convert the laser-microcavity, relative-frequency noise to power fluctuations in the transmit-
ted signal. Therefore the power spectral density of the cavity transmission,
S
P
(
Ω
)
, is related to
the power spectral density of the laser-microcavity frequency noise
S
lc
Δ
ω
(
Ω
)
as follows:
S
P
(
Ω
)=
P
2
in
H
(
Ω
)
S
lc
Δ
ω
(
Ω
)
(20)
where
P
in
is the input power and
H
(
Ω
)
is the cavity frequency discrimination function. Here
we neglect the laser intensity noise contributions to the output power fluctuations.
H
(
Ω
)
has
the following form [12],
H
(
Ω
)=
κ
2
ex
[
Δ
2
+
κ
2
/
4
]
2
4
Δ
2
(
κ
2
i
+
Ω
2
)
[(
Δ
+
Ω
)
2
+
κ
2
/
4
][(
Δ
−
Ω
)
2
+
κ
2
/
4
]
(21)
For Fourier frequencies much less than the cavity linewidth,
H
(
Ω
)
≈
H
o
=
4
κ
2
ex
κ
2
i
Δ
2
[
Δ
2
+
κ
2
/
4
]
4
.
In the measurement, a resonance ina2mmdisk resonator [5] with loaded linewidth of
4.1 MHz (Q = 47 million) at critical coupling is thermo-optically locked using an external
cavity diode laser (ECDL). A series of input power levels (from 25
μ
W to 2 mW) are ap-
plied with the laser detuned to the half-linewidth point (i.e.,
Δ
=
κ
/
2). The measured laser-
microcavity, relative-frequency-noise spectra are given in Fig. 2(a). Note that
S
lc
Δ
ν
≡
S
lc
Δ
ω
/
4
π
2
#209930 - $15.00 USD
Received 10 Apr 2014; revised 30 May 2014; accepted 30 May 2014; published 5 Jun 2014
(C)
2014
OSA
16
June
2014
| Vol.
22,
No.
12
| DOI:10.1364/OE.22.014559
| OPTICS
EXPRESS
14563
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
−70
−60
−50
−40
−30
−20
−10
0
10
Frequency (Hz)
Measured R( f )(dB)
P
in
25uW
P
in
100uW
P
in
500uW
P
in
2mW
a
b
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
−2
10
−1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
Frequency (Hz)
S
Δν
(f) (Hz
2
/Hz)
Free running
Thermolock 25uW
Thermolock 100uW
Thermolock 500uW
Thermolock 2mW
Noise floor
lc
Fig. 2. (a) The laser-microcavity relative-frequency-noise spectra are measured for a 2 mm
disk resonator with loaded linewidth 4.10 MHz under a series of input power levels: 25
μ
W
(red curve), 100
μ
W (green curve), 500
μ
W (cyan curve) and 2 mW (blue curve). For com-
parison, the free-running laser frequency noise is also shown (black curve), measured using
a Mach-Zehnder interferometer [5]. The magenta curve is the background noise when the
laser is off-resonance and contains the laser relative intensity noise (RIN) and shot noise
normalized by the collected power and transfer function
H
. Background noise (magenta
curve) rises above 4 MHz is a result of the shot noise divided by the cavity transfer func-
tion. Note that the spurious noise at low frequencies (e.g. at 10 Hz) is believed to result
from corresponding peaks in the frequency noise (e.g. mechanical noise) of the pump laser.
(b) TOL laser-microcavity relative-frequency-noise transfer function,
R
, is plotted at a se-
ries of input power levels given in (a). The black-dashed curves are calculations based on
Eq. (16) using the following fitting parameters
γ
th
/2
π
= 128 Hz and
α
/2
π
= 13.5 kHz/mW.
TOL-induced laser-microcavity relative-frequency-noise suppression increases with the in-
put power as does the TOL bandwidth.
#209930 - $15.00 USD
Received 10 Apr 2014; revised 30 May 2014; accepted 30 May 2014; published 5 Jun 2014
(C)
2014
OSA
16
June
2014
| Vol.
22,
No.
12
| DOI:10.1364/OE.22.014559
| OPTICS
EXPRESS
14564