We study the optomechanical design introduced by M. Li et al. [Phys. Rev.
Lett. {\bf 103}, 223901 (2009)], which is very effective for investigation of
the effects of reactive coupling. We show the normal mode splitting which is
due solely to reactive coupling rather than due to dispersive coupling. We
suggest feeding the waveguide with a pump field along with a probe field and
scanning the output probe for evidence of reactive-coupling-induced normal mode
splitting.Comment: 4 pages,6 figure

Agarwal, G. S.

Huang, Sumei

English

arXiv

Sumei Huang

G. S. Agarwal

Crossref

Reactive-coupling-induced normal mode splittings in microdisk resonators coupled to waveguides

We study the optomechanical design introduced by M. Li et al. [Phys. Rev. Lett. 103, 223901 (2009)], which is very effective for investigation of the effects of reactive coupling. We show the normal mode splitting that is due solely to reactive coupling rather than due to dispersive coupling. We suggest feeding the waveguide with a pump field along with a probe field and scanning the output probe for evidence of reactive-coupling-induced normal mode splitting

Publications of the IAS Fellows

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Reactive-Coupling-Induced Normal Mode Splittings in Microdisk Resonators Coupled
to Waveguides
Sumei Huang and G. S. Agarwal
Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA
(Dated: May 11, 2010)
We study the optomechanical design introduced by M. Li et al. [Phys. Rev. Lett. 103, 223901
(2009)], which is very effective for investigations of the effects of reactive coupling. We show the
normal mode splitting that is due solely to reactive coupling rather than due to dispersive coupling.
We suggest feeding the waveguide with a pump field along with a probe field and scanning the
output probe for evidence of reactive-coupling-induced normal mode splitting.
PACS numbers: 42.50.Wk, 42.65.Dr, 42.65.Sf, 42.82.Et
I. INTRODUCTION
In a recent paper Li et al. [1] presented a new de-
sign for an optomechanical system that consists of a mi-
crodisk resonator coupled to a waveguide. This design
has several attractive features. Besides its universality, it
enables one to study the reactive effects [1, 2] in optome-
chanical coupling. The origin of the reactive coupling is
well explained in Ref. [3]. Its origin lies in the mechani-
cal motion dependence of the extrinsic losses of the disk
resonator. Further phase-dependent gradient forces lead
to reactive coupling. Li et al. have also argued that this
design is more effective in achieving cooling of the sys-
tem to its ground state. While cooling is desirable for
studying quantum effects at the macroscopic scale [4–9],
we examine other possibilities, which do not depend on
the cooling of the system, to investigate the effects arising
from strong reactive coupling. Since optomechanical cou-
pling effects are intrinsically nonlinear, we examine the
nonlinear response of the microdisk resonator to pump
probe fields. We report reactive-coupling-induced nor-
mal mode splitting. Note that in previous works [10–13]
on normal mode splitting in optomechanical devices, only
dispersive coupling was used. In this paper, we report on
normal mode splitting due to reactive effects.
The paper is organized as follows. In Sec. II, the phys-
ical system is introduced and the time evolutions of the
expectation values of the system operators are given and
solved. In Sec. III, the expectation value of the output
fields is calculated, and the nonlinear susceptibilities for
Stokes and anti-Stokes processes are obtained. In Sec.
IV, we discuss normal mode splitting in output fields
with or without reactive coupling. We find that there is
no normal mode splitting in output fields in the absence
of reactive coupling. However, normal mode splitting oc-
curs in output fields in the presence of reactive coupling.
II. MODEL
We consider the system shown in Fig. 1, in which
a microdisk cavity is coupled to a freestanding waveg-
uide. A strong pump field with frequency ωl and a weak
Stokes field with frequency ωs enter the system through
the waveguide. The waveguide will move along the y di-
rection under the action of the optical force exerted by
the photons from the cavity. Further, considering the
dispersive coupling and reactive coupling between the
waveguide and the cavity, displacement q of the waveg-
uide from its equilibrium position will change the reso-
nant frequency of the cavity field and the cavity decay
rate, represented by ωc(q) and κe(q), respectively.
FIG. 1: (Color online) Sketch of the studied system (from
Ref.[1]). The microdisk cavity is driven by a pump field and a
Stokes field. The nonlinearity of the interaction also generates
anti-Stokes field.
In a rotating frame at pump frequency ωl, the Hamil-
tonian of the system is given by [1]
H = ~[ωc(q)− ωl]c†c+ p
2
2m
+
1
2
mω2mq
2
+~L
c
n˜g(ωlε
2
l + ωs|εs|2) + i~
√
2κe(q)εl(c
† − c)
+i~
√
2κe(q)(εse
−iδtc† − ε∗seiδtc).
(1)
The first term is the energy of the cavity field, whose
annihilation (creation) operators are denoted c(c†). The
second and third terms are the energy of the waveguide
with mass m, frequency ωm, and momentum operator
p. The fourth term gives the interactions between the
waveguide and the incident fields (the pump field and
the Stokes field), L is the length of the waveguide, c is
2the speed of light in vacuum, n˜g is the group index of the
waveguide optical mode [14], εl and |εs| are the ampli-
tudes of the pump field and the Stokes field, respectively,
and they are related to their corresponding power ℘l and
℘s by εl =
√
℘l
~ωl
and |εs| =
√
℘s
~ωs
. The latter two terms
describe the coupling of the cavity field to the pump field
and the Stokes field, respectively. And δ = ωs−ωl is the
detuning between the Stokes field and the pump field. We
would study the physical effects by scanning the Stokes
laser.
For a small displacement q, ωc(q) and κe(q) can be
expanded to the first order of q,
ωc(q) ≈ ωc + qχ,
κe(q) ≈ κe + qκom,
(2)
thus the quantities χ and κom describe the cavity-
waveguide dispersive and reactive coupling strength, re-
spectively. Further, note that the photons in the cavity
can leak out of the cavity by an intrinsic damping rate
κi of the cavity and by a rate of κe(q) due to the reac-
tive coupling between the waveguide and the cavity. In
addition, the velocity of the waveguide is damped at a
rate of γm. Applying the Heisenberg equation of motion
and adding the damping terms, the time evolutions of
the expectation values (〈q〉, 〈p〉, and 〈c〉) for the system
can be expressed as
〈q˙〉 = 〈p〉
m
,
〈p˙〉 = −mω2m〈q〉 − ~χ〈c†〉〈c〉 − 2~
κom√
κ
×Im[(εl + ε∗seiδt)〈c〉]− γm〈p〉,
〈c˙〉 = −[κ+ 〈q〉κom + i(ωc − ωl + 〈q〉χ)]〈c〉
+
√
κ[1 + 〈q〉κom
κ
](εl + εse
−iδt),
(3)
where we have used the mean field assumption 〈qc〉 =
〈q〉〈c〉, expanded κe(q) to the first order of q, and assumed
κe ≈ κi ≈ κ/2, where κ is the half-linewidth of the cavity
field. It should be noted that the steady-state solution of
Eq. (3) contains an infinite number of frequencies. Since
the Stokes field εs is much weaker than the pump field
εl, the steady-state solution of Eq. (3) can be simplified
to first order in εs only. We find that in the limit t→∞,
each 〈q〉,〈p〉, and 〈c〉 has the form
〈s〉 = s0 + s+εse−iδt + s−ε∗seiδt, (4)
where s stands for any of the three quantities q, p, and c.
Thus the expectation values (〈q〉, 〈p〉, and 〈c〉) oscillate
at three frequencies (ωl, ωs, and 2ωl − ωs). Substituting
Eq. (4) into Eq. (3), ignoring those terms containing the
small quantities ε2s, ε
∗2
s , |εs|2, and equating coefficients
of terms with the same frequency, respectively, we obtain
the following results
c0 =
Aεl
κ+ q0κom + i∆
,
q0 = − ~
mω2m
[χ|c0|2 + iκom√
κ
εl(c
∗
0 − c0)],
c+ =
1
d(δ)
[A(BE + FJ)− i~κom√
κ
c∗0BF
∗],
c− =
F ∗
d∗(δ)
(−AJ + i~κom√
κ
c0V ),
q+ =
B
d(δ)
(−AJ∗ − i~κom√
κ
c∗0V
∗),
q− = (q+)
∗,
(5)
where
∆ = ωc − ωl + χq0, (6)
d(δ) = V ∗(BE + FJ) +BF ∗J∗, (7)
and A =
√
κ(1 +
κom
κ
q0), B = κ + q0κom − i(∆ + δ),
E = m(ω2m − δ2 − iγmδ), F = −c∗0(κom − iχ) +
κom√
κ
εl,
J = χ~c0 + i~
κom√
κ
εl, V = κ + q0κom − i(∆ − δ). The
approach used in this paper is similar to our earlier work
[13] which dealt with optomechanical systems with dis-
persive coupling only.
III. OUTPUT FIELDS
To investigate the normal mode splitting of the output
fields, we need to calculate their expectation value. It
can be obtained by using the input-output relation [15]
〈cout〉 =
√
2κe(q)〈c〉. If we write 〈cout〉 as
〈cout〉 = cl + εse−iδtcs + ε∗seiδtcas, (8)
where cl is the response at the pump frequency ωl, cs is
the response at the Stokes frequency ωs, and cas is the
field generated at the new anti-Stokes frequency 2ωl−ωs.
Then we have
cl =
√
κ(1 +
κom
κ
q0)c0,
cs =
κom√
κ
q+c0 +
√
κ(1 +
κom
κ
q0)c+,
cas =
κom√
κ
q−c0 +
√
κ(1 +
κom
κ
q0)c−.
(9)
30 200 400 600 800 1000
0.7
0.8
0.9
1.0
1.1
1.2
1.3
0.7
0.8
0.9
1.0
1.1
1.2
1.3
Ãl HΜWL
R
eH
∆
L
Ω
m
FIG. 2: (Color online) The real roots of d(δ) in the domain
Re(δ) > 0 as a function of the pump power ℘l for κom = 0
(dotted curve) and κom = −2pi× 26.6 MHz/nm (solid curve).
0 200 400 600 800 1000
0.00
0.05
0.10
0.15
0.20
0.25
0.00
0.05
0.10
0.15
0.20
0.25
Ãl HΜWL
-
Im
H∆
L
Ω
m
FIG. 3: (Color online) Imaginary parts of the roots of d(δ) as
a function of the pump power ℘l for κom = 0 (dotted curve)
and κom = −2pi × 26.6 MHz/nm (solid curve).
Furthermore, whether there is normal mode splitting in
the output fields is determined by the roots of the de-
nominator d(δ) of cs. Here we examine the roots of d(δ)
given by Eq. (7) numerically.
The response of the system is expected to be especially
significant if we choose ωs corresponding to a sideband
ωs = ωl ± ωm or ωs = ωl ± ∆, so we consider the case
∆ = ωm. The other parameters are chosen from a recent
experiment focusing on the effect of the reactive force
on the waveguide [1]: the wavelength of the laser λ =
2pic/ωl = 1564.25 nm, χ = 2pi × 2 MHz/nm, m = 2
pg (density of the silicon waveguide, 2.33 g/cm3; length,
10 µm; width, 300 nm; height, 300 nm), κ = 0.2ωm,
ωm = 2pi×25.45 MHz, and the mechanical quality factor
Q = ωm/γm = 5000. In the following, we work in the
stable regime of the system.
Figure 2 shows the variation of the real parts of the
roots of d(δ) in the domain Re(δ) > 0 with increasing
pump power for no reactive coupling, κom = 0, and for
κom = −2pi × 26.6 MHz/nm. For κom = 0, the interac-
tion of the waveguide with the cavity is purely dispersive;
the cavity decay rate does not depend on the displace-
ment of the waveguide. In this case, the real parts of the
roots of d(δ) always have two equal values with increas-
ing pump power. Thus there is no splitting because the
dispersive coupling is not strong enough. However, for
κom = −2pi × 26.6 MHz/nm, the system has both dis-
persive and reactive couplings, the cavity decay rate de-
pends on the displacement of the waveguide, and the real
parts of the roots of d(δ) will change from two equal val-
ues to two different values with increasing pump power.
And the difference between two real parts of the roots
of d(δ) in the domain Re(δ) > 0 is increased with in-
creasing pump power. Therefore, the reactive coupling
between the waveguide and the cavity can result in nor-
mal mode splitting of the output fields, and the peak
separation becomes larger with increasing pump power.
Figure 3 shows the variation of the imaginary parts of
the roots of d(δ) with increasing pump power for zero
reactive coupling κom = 0 and nonzero reactive coupling
κom = −2pi×26.6 MHz/nm. For κom = 0, the imaginary
parts of the roots of d(δ) do not change with increasing
pump power. However, for κom = −2pi × 26.6 MHz, the
imaginary parts of the roots of d(δ) change with increas-
ing pump power. We thus conclude that for the present
microdisk resonator coupled to a waveguide the normal
mode splitting is solely due to the reactive coupling.
0.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
∆Ωm
Υ
s
Ãl=200 ΜW
Ãl=20 ΜW
FIG. 4: (Color online) The lower two curves show the normal-
ized quadrature vs as a function of the normalized detuning
between the Stokes field and the pump field, δ/ωm for κom = 0
(dotted curve) and κom = −2pi × 26.6 MHz/nm (solid curve)
for pump power ℘l = 20 µW. The upper two curves give the
normalized quadrature vs+1.5 for pump power ℘l = 200 µW.
IV. NORMAL MODE SPLITTING IN OUTPUT
FIELDS
We now discuss how the output fields depend on the
behavior of the roots of d(δ). For convenience, we nor-
malize all quantities to the input Stokes power ℘s. As-
suming that εs is real, we express the output power at the
Stokes frequency ωs in terms of the input Stokes power
Gs =
~ωs|εscs|2
℘s
= |cs|2. (10)
Further, we introduce the two quadratures of the Stokes
component of the output fields by υs =
cs + c
∗
s
2
and
40.0 0.5 1.0 1.5 2.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
∆Ωm
G
s
Ãl=200 ΜW
Ãl=20 ΜW
FIG. 5: (Color online) The lower two curves show the normal-
ized output power Gs as a function of the normalized detuning
between the Stokes field and the pump field, δ/ωm for κom = 0
(dotted curve) and κom = −2pi × 26.6 MHz/nm (solid curve)
for pump power ℘l = 20 µW. The upper two curves give the
normalized output power Gs+1.5 for pump power ℘l = 200
µW.
0.0 0.5 1.0 1.5 2.0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
∆Ωm
G
as
Ãl=200 ΜW
Ãl=20 ΜW
FIG. 6: (Color online) The lower two curves show the nor-
malized output power Gas as a function of the normalized
detuning between the Stokes field and the pump field, δ/ωm
for κom = 0 (dotted curve) and κom = −2pi × 26.6 MHz/nm
(solid curve) for pump power ℘l = 20 µW. The upper two
curves give the normalized output power Gas+0.15 for pump
power ℘l = 200 µW.
υ˜s =
cs − c∗s
2i
. One can measure either the quadratures
of the output by homodyne techniques or the intensity
of the output. For brevity, we only show υs and Gs as a
function of the normalized detuning between the Stokes
field and the pump field δ/ωm for this model, without re-
active coupling (κom=0) and with it (κom = −2pi × 26.6
MHz/nm), for different pump powers in Figs. 4–5. For
κom=0, it is found that υs has a Lorentzian lineshape
corresponding to the absorptive behavior. Note that υs
and Gs exhibit no splitting when κom=0. However, for
κom = −2pi×26.6 MHz/nm, it is clearly seen that normal
mode splitting appears in υs and Gs. Therefore reactive
coupling can lead to the appearance of normal mode split-
ting in the output Stokes field. And the peak separation
increases with increasing pump power [16]. The dip at
the line center exhibits power broadening. We also find
that the Stokes field can be amplified by the stimulated
process. Obviously the maximum gain Gs for the Stokes
field depends on the system parameters. For a pump
power ℘l = 200 µW, the maximum gain for the Stokes
field is about 1.3.
Note that the nonlinear nature of the reactive cou-
pling generates anti-Stokes radiation. In a similar way,
we define a normalized output power at the anti-Stokes
frequency 2ωl − ωs as
Gas =
~(2ωl − ωs)|εscas|2
℘s
= |cas|2. (11)
The plots of Gas versus the normalized detuning be-
tween the Stokes field and the pump field δ/ωm for this
model, without reactive coupling (κom=0) and with it
(κom = −2pi × 26.6 MHz/nm), for different pump pow-
ers are presented in Fig. 6. We can see that Gas ≈ 0
for κom=0. The reason is that the dispersive coupling
constant χ is too small. However, for κom = −2pi × 26.6
MHz/nm, Gas is not equal to zero. This shows that the
optomechanical system can generate an anti-Stokes field
with frequency (2ωl − ωs) due to the reactive coupling.
For pump power ℘l = 200 µW, the maximum gain de-
fined with reference to the input Stokes power for the
anti-Stokes field is about 0.1.
V. CONCLUSIONS
In conclusion, we have observed normal mode split-
ting of output fields due to reactive coupling between the
waveguide and the cavity. Meanwhile, the separation of
the peaks increases for larger pump powers. Further, the
reactive coupling can also cause four-wave mixing, which
creates an anti-Stokes component generated by the op-
tomechanical system.
We gratefully acknowledge support from NSF Grant
No. PHYS 0653494.
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Reactive-Coupling-Induced Normal Mode Splittings in Microdisk Resonators Coupled to Waveguides

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