Compensation of thermal nonlinearity effect in
optical resonators
Ivan Grudinin,
1,2
Hansuek Lee,
1,2
Tong Chen,
1
and Kerry Vahala
1,*
1
Department of Applied Physics, Caltech, 1200
E. California Blvd., Pasadena, 91125 CA, USA
2
Equal contributions
*vahala@caltech.edu
Abstract:
Thermal nonlinearity is known to cause bistability in Whispering
Gallery Mode (WGM) resonators and to destabilize the red slope of the
Lorentzian resonant curve. We demonstr
ate an optical technique that allows
compensation of the thermal effect a
nd forces the resonances to appear
linear with both red and blue slopes stable.
©2011 Optical Soci
ety of America
OCIS codes:
(140.3320) Laser cooling; (140.4780) Op
tical resonators; (230.1150) All-optical
devices; (140.3945) Microcavities; (
190.4870) Photothermal effects
References and links
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V. S. Ilchenko, “Nonstationary nonlinear effects in optical
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1. Introduction
Whispering Gallery Mode (WGM) resonators have numerous applications in science and
technology [1], including cavity QED [2,3], cavity optomechanics [4] nonlinear optics [5–7]
and biosensing [8]. Thermal nonlinearity (caused by light absorption in the WGM) is present
in virtually every WGMR and is known to cause bistability [9] at high circulating power
levels. Importantly, at room temperature it renders the lower frequency slope (the “red” slope)
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of the WGM resonance unstable in silica resonators [10]. Observation of the actual line shape
function must therefore occur at reduced power le
vels. Attempts have been made to cancel or
reduce the thermal effect [11,12] at the expense of the optical quality factor. In this
contribution we demonstrate a technique that a
llows one to cancel thermal nonlinearity in a
WGM of choice using another WG mode in
the same resonator as a stabilizer. We
demonstrate the method by cooling a mechanical mode of the resonator through red detuning
of a pump wave.
2. Theoretical model
Thermal shift of the optical resonator eigenfrequency is caused by the absorption of coupled
laser power and subsequent heating. In WGM resonators the frequency shift
Δω
is caused
primarily by temperature dependence of
refractive index n and thermal expansion
α
T
:
1
T
n
TT
nT
(1)
However, if another WGM in the same resonato
r is excited at a diffe
rent wavelength by a
second laser and if this laser is thermally locked [10] on the stable “blue” slope, then the
optical pumping of the original WGM can be stabilized. Before describing experimental
results, we first detail the numerics used to study the stabilization mechanism. A simplified
model is also described in section A. The system consists of a cavity with two WGM
resonances at different wavelengths, but sharing approximately the same volume. A system of
coupled equations similar to those in [13,14] is used to analyze the thermal behavior. Table 1
lists the variables and definitions. Subscripts “
a”
and “
b”
denote the parameters of the
corresponding resonator modes.
Table 1. Parameters and variables used in
computation of resonance curves in Fig. 2
Parameter, notation
Value
Optical quality factor, Q
Q
a
= 1.4 × 10
7
, Q
b
= 7 × 10
6
Density,
ρ
2.2 g/c
m
3
Thermal conductivity,
k
1.4 × 10
5
erg/(cm K s)
Heat capacity, C
6.7 × 10
6
erg/(g K)
Thermal diffusivity, D
/)(
Dk C
=
9.5 × 10
3
cm
2
/s
Third order susceptibility,
χ
(
3
)
1.3 × 10
14
cm
3
/erg (esu) [15]
Absorption coefficient,
α
5 × 10
4
cm
1
Thermal expansion and thermorefractivity,
β
8.83 × 10
6
K
1
Refractive index, n
1.444
Laser wavelength,
λ
1540, 1450 n
m
Resonator major radius,
R
29
μ
m
Speed of light in vacuum, c
29979245800 cm/s
WGM eigenfrequencies,
ω
a,b
, Variables
WGM volumes,
V
a,b
. (
l,m
are WGM indices)
(11/ 6 )
( 7 / 6 )
3.9
2
/
() 1
()
Vl
Rcn
m
WGM optical loss rates,
γ
a,b
/(2 )
Q
Pump power from laser 1 and 2, W
1,2
erg/s
Generalized force created by lasers 1 and 2
2
1, 2
,
,
1, 2
2/( )
ab
ab
FWnVQ
Kerr and thermal nonlinearity, μ and
ν
(3)
2
2/
n
and
/ (4
)
nc
C
Thermal relaxation rate of the resonato
r
2
/( /4)
DR
We describe the electric field of the WGMs using the rotating wave approximation:
0
(,)
() () ,
it
Ert
atE re
(2)
where the field distribution of a
mode is normalized so that
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2
0
1.
Edr
(3)
We introduce the equations describing the dynamics of the slowly varying amplitudes
a
and
b
in cgs units as
11
2
2
2
2
22
([ (1
)])
(
[
(1
)])
.
()
aa
bb
aa
i
a
iF
bb
i
b
iF
ab
(4)
Here
2
0
0
)(
()
r
TEdr
T
is the relative temperature averaged over the volume of the
modes (assumed equal for the two WGMs). The WGM volume formula given in Table 1 is an
approximation computed for a sphere, and numerical simulations or other approximations
may be used for spheroidal resonators. The therma
l nonlinearity observed in the experiment is
a result of mutual dynamics of the laser’s fr
equency scan and laser
induced eigenfrequency
5
,
c
y
change in the WGM. The effective time
constant of the thermal nonlinearity
depends on the laser scan rate. Static behavior of the resonator corresponds to the longest time
constant describing thermal relaxation of the whole resonator.
2.1 Solution of the nonlinear system
To deal with the complex numbers in Eqs. (4), field amplitudes are defined as
12
aa
ai
and
12
bb
bi
. We look for the solutions in which
1212
,,,
b
aa
b
are real:
21
212
2
11 21
2
21
1
2
11 22
2
2
22
2
()
)
()
)
()
)
()
(1
(1
(1
(1
)
()
aa
b
b
aa
b
b
aa
aa
b
aa
aaF
bb
bF
bb
ab
b
b
(5)
In order to accommodate both very large and very small quantities in a numerical solver
we use non-dimensional variables. The following substitutions are made in Eqs. (5):
11
,
c
aya
22
,
c
aya
13
,
c
byb
24
,
c
byb
,
c
tt
and
,1,2,
ab
ab
where
,
c
b
aa
c
a
ab
,
a
c
a
, and
1
c
a
t
. With these substitutions, the
equations contain only dimensionless values which are close to unity in magnitude
(derivatives here are with respect to
τ
):
23
112
21225
221
1 151
23
33
4
45
434
32
534
2
32
3
12
3
44
3
3
3
2
2
2
55
12
3
/
//
// /
// /
/
/(
/( (
))
aa
aa
a
ba
b a
a
ba
b
ba
b a
aba
aa
yyy
y
yyy
yy yF
yy
y
y
yy
yyyy
yy
yy
y
y
yy
y
y
yy
y
yyF
yy
2
4
.
)/( )
aab
y
(4
)
We set all time derivatives to zero and find the static solutions of the system (4
) using the
numerical solver Maple. The power circulating in resonator mode
”a”, for example, can be estimated as
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222
12
7
2
)10]
(
)
[.
(2
a
cir
c
c
ncV
Pya
Watt
y
R
(6)
We also check the static solutions for stability against minor deviations of variables
y
i
in
the Lyapunov sense. Stability analysis is performed as previously described in [13]. To model
the experimental outcomes, we fix the frequency of the first laser
ω
1
and scan the frequency of
the second laser to observe the stabilization effect
, as shown in the next section. We disregard
the cross-mode modulation via the Kerr effect as the thermal effect is the dominant
nonlinearity in our case.
3. Experimental verification
We use two New Focus velocity lasers to
probe the resonator. The stabilizing laser (
ω
1
,
λ
1
1550nm
) excites the WGM at
λ
= 1543.2 nm and the probing laser (
ω
2
,
λ
1
1450nm
)
operates at
λ
= 1433.3 nm. We use a WDM coupler to combine the two lasers and to separate
the optical signals passing through the resonator. A tapered fiber coupler [16,17] is used to
excite the WGMs of an ellipsoidal resonator with
a major radius of 29 micrometers fabricated
on a silicon chip. Experimental setup diagram is shown on Fig. 1.
Fig. 1. Schematic of the experimental setup.
Fig. 2. A) experimental frequency scans of
a WGM resonance at 1450 nm without (red) and
with stabilization (blue). B) solutions of the system (4). WGM
resonance with thermal effect
(red dots show unstable solutions). Solutions for
the same mode when the stabilizing laser is
activated constitute a nearly Lo
rentzian lineshape. Note the fr
equency shift of the stabilized
resonance caused by heating from the stabilizing WGM.
During the experiment we first scan the 1450 nm laser alone over the resonance and record
the resonance with the thermal non
linearity (red trace on the left panel of Fig. 2). The shape of
this resonance is slightly different from the static solution due to dynamic effects. We then
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activate the stabilizing laser at 1550 nm and thermally lock it to the blue slope of the nearby
resonance. Heating from this laser causes the 1450 nm resonance to shift by a value that
depends on the pump power. With the stabilizing laser enabled, the 1450 nm WGM exhibits a
more Lorentzian lineshape (blue trace
on the left panel of Fig. 2).
The vertical axis of the right panel give
s the calculated circulating power for the
experimental conditions: coupled power of 300
μ
W at 1550 nm, and 130
μ
W at 1450 nm; and
loaded quality factors of 1.4 × 10
7
and 7 × 10
6
, respectively.
If the Q of both modes is similar, a much
higher coupled power (r
elative to the probe
power of the stabilized mode) is required. In we
ll fabricated WGMs, especially in spheroidal
ones, most modes have high optical Q and it may be difficult to find a pair of modes with
different Q. Nonetheless, the
difference in Q can still be achieved by using an asymmetric
coupling technique, wherein distin
ct transverse WGMs (different number of field maxima in
vertical direction) are used. Such modes have maxima shifted in the vertical direction,
enabling a strong difference in loaded Q of the two modes by a single taper waveguide. The
modes in our experiment were, in fact, of a different order and the higher Q mode (3 maxima
in vertical direction, l-m = 2) was more weakly loaded than the stabilized mode (2 maxima, l-
m = 1). As shown in Fig. 2, the numerical model verifies the experimental results with good
accuracy. The thermal relaxation rate was chosen
to correspond to a quarter of resonator
radius R to match experimental results.
We further tested the stability of the 1450nm mode by manually tuning the probing laser
to the blue and red slopes. The induced stability
allowed us to apply optomechanical cooling
and amplification to a mechanical mode present in our optical resonator (symmetrically on
both slopes of the optical resonance). Doing so in the unresolved sideband regime we obtained
Fig. 3. Optical cooling and
amplification of a 70 MHz mech
anical mode of the optical
resonator. Broadening and narrowing of the mechanical spectrum occurs due to
optomechanical cooling and amplification in
the unresolved sideband regime. Colored lines
represent Lorentzian fits of the experimental
ly recorded spectra. Inset shows an optical
photograph of the resonator near the fiber coupler.
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modest but measurable cooling and amplification, as shown in Fig. 3. It is important to note
that cooling (i.e., stable operation on the red slope) was not possible in the thermally
unstabilized system.
To further explain the data presented in Fig. 3, laser light passing through the optical
resonator is modulated by cavity vibrations which impose optical sidebands. These are
demodulated by a photodetector, which provides an electronic signal with signatures of the
mechanical resonances. This signal is then di
splayed by the electrical spectrum analyzer and
the linewidth of the mechanical oscillator can be analyzed [4]. The spectra presented here
were obtained by detecting the optical signal at 1550 nm that was thermally locked to the
cavity resonance.
4. Conclusion
We have demonstrated an all-optical technique
that makes it possible to counteract thermal
nonlinearity in WGM resonators and makes the
red slope of the resonant curve accessible. To
demonstrate the technique we have applied it to optical cooling of the mechanical mode of an
optomechanical resonator by pumping the resonator mode in the red detuned regime. This
application to optomechanical cooling should
be considered only illustrative as there are
potential problems in applying the method to achieve large amounts of cooling. Among these
is the onset of parasitic oscillation of mechanical modes as well as heating of the resonator by
the stabilizing pump wave. Nonetheless, the method might find us
e in applications wherein it
is necessary to observe stable resonances at higher power levels.
Appendix A. Analytical model
To understand the underlying stabilization mechanism, we develop a simple nonlinear
dynamics model. In the case of a single WGM
excitation, cavity heating by the pump laser
depends on the pump power, the optical coupling efficiency, and detuning from the resonance
frequency. Based on the conservation of energy, the net heating of the cavity is the laser pump
heating (
in
q
) minus the thermal dissipation (
out
q
) [10]:
2
2
,0
1
((1))/1
in
h
b
u
b
ot
Cqq W
K
. (A1)
Here
2
is the laser frequency;
,0
(1
)
bb
is the resonance frequency; and we have
defined the effective laser power
tot
h
abs
W
Q
W
Q
, where
abs
Q
is the abso
rption quality factor of
the cavity and
η
is the coupling efficiency. The st
ability of WGM can be inferred by
investigating the stability of the steady solution to Eq. (A1). The rearrangement of Eq. (A1)
yields
2
2
,0
2
22 3
2
2
2
2
,0
2
,0
,0
2
2
,0
),
(1
(
,
)
2()
()
bb
h
h
bbb
bbb
C
fW
K
W
K
(A2)
The steady state solutions for
0
and their stability can be determined from the phase
diagram (Fig. 4). For blue detuning of the pump laser
2
()
b
, the monotonic form of
2
,
,)
(
h
f
W
leads to a single, stable, steady state; whereas on red side
2
()
b
, thermal
bistability may arise at sufficient pump power (
h
W
). The bistability would emerge as the
saddle-node bifurcation when
h
W
K
is above a certain threshold (i.e. large pump power or small
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Fig. 4. Left panel: Phase diagram of temperat
ure change of the cavity upon application of a
blue detuned pump. There is always a single, stable steady-state solution. Right panel: Phase
diagram of temperature change of the cavity upon using a red detuned pump. The thermal
bistability may arise at sufficient pump power
. The excitation of a stabilizing mode can
compensate the nonlinearity via a saddle-node bifurcation.
thermal dissipation). In other words, the
increase
of thermal conductivity or the
decrease
of
pump power would eliminate this bistability
. In the present syst
em, the blue-detuned
excitation of another (stabilizing) WGM compensates the thermal nonlin
earity by creating an
“effective” enhancement to ther
mal conductivity. The stabilizing mode adds a second term to
the dynamical equation as follows:
,
2
2
,,
1
(()
().
()/1
)
hs
hc
b
s
cc
b
CW
K
W
WK
W
(A3)
Here, the effective laser pow
er for the stabilizing mode,
s
W
is defined in the same fashion
as
h
W
.
2
,1
/)1
(/
sc
s
a
a
WW
and
2
,2
/{[(
)/ ]
1}
hc
h
b
b
WW
are the coupled
power from the stabilizing and or
iginal pump mode. We are intere
sted in the stabilizing effect
of this additional term
,
s
c
W
and explore stability of th
e steady state solutions
s
of (A3)
against small deviations. With a substitution of
s
T
into Eq. (A3) we get
,,
,
,
hc
sc
hc
sc
a
a
WWW
C
W
TTTTT
KKT
(A4)
When the stabilizing mode is excited on the blue side, the positive
,
s
c
a
W
will effectively
increase the thermal conductivity and eliminate the bistability via a sadd
le-node bifurcation.
For the pump mode,
,
hc
W
=
,
hc
b
b
W
is negative on the blue
slope and the solution is
always stable. On the red slope,
,
hc
W
is positive and th
e coefficient of
T
on the right hand
side of (A4) must be nega
tive to ensure the stability, l
eading to stability condition:
,,, ,
hc
sc
hc
sc
ab
a
ab
a
WWW W
KK
(A5)
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For optimal pump detuning of the
stabilizing mode (maximum value of
,
s
c
a
W
), we find
,
33
4
sc
s
a
aa
opt
W
QW
, and the right hand side of Eq. (A5) has to be greater than the
maximum value of
a
,
b
hc
W
. We finally get the stability condition:
33
33
44
s
b
ha
K
WW
QQ
Acknowledgments
We would like to acknowledge the support from DARPA.
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