Download Whispering gallery modes of microspheres in the presence of a

JOURNAL OF APPLIED PHYSICS 107, 103105 共2010兲
Whispering gallery modes of microspheres in the presence of a changing
surrounding medium: A new ray-tracing analysis and sensor
experiment
Tindaro Ioppolo,1,a兲 Nirod Das,2 and M. Volkan Ötügen1
1
Department of Mechanical Engineering, Southern Methodist University, Dallas, Texas 75205, USA
Department of Electrical and Computer Engineering, Polytechnic Institute of New York University, Brooklyn,
New York 11201, USA
2
共Received 9 November 2009; accepted 7 April 2010; published online 20 May 2010兲
A simple plane wave, ray-tracing approach was used to derive approximate equations for the
dielectric microsphere whispering gallery mode 共WGM兲 resonant wavenumber and quality factor, as
dependent on the surrounding medium’s refractive index. These equations are then used to
determine the feasibility of a micro-optical sensor for species concentration. Results indicate that the
WGMs are not sensitive enough to refractive index changes in the case of gas media. However, they
can be sufficiently sensitive for measurements in liquids. Experiments were carried out to validate
the analysis and to provide an assessment of this sensor concept. © 2010 American Institute of
Physics. 关doi:10.1063/1.3425790兴
2␲Rn1 = l␭,
I. INTRODUCTION
Micro-optical sensor concepts based on the so-called
whispering gallery mode 共WGM兲 shifts in spherical dielectric resonators were recently demonstrated for temperature,1
force,2,3 pressure,4 and electric field.5 In these studies, a
single mode optical fiber is used to couple tunable laser light
into the sphere. The WGMs are observed as sharp dips in the
transmission spectrum through the fiber. The observed linewidth, ␦␭, of these dips is related to the quality factor, Q
= ␭ / ␦␭. A small perturbation in the shape, size, or the index
of refraction of the microsphere causes a shift in the resonance positions allowing for the precise measurement of the
external condition causing the change in the morphology of
the sphere. In addition to the morphology of the sphere, the
WGMs are affected also by changes in the refractive index of
the surrounding medium. Thus, a change in the composition
of the surrounding medium, if sufficiently large, will induce
a detectable shift in the WGMs. This effect can be used, in
principle, to monitor species concentration changes in the
medium surrounding a dielectric microcavity. Krioukov et
al.,6 recently proposed a species concentration sensor using
this approach. They monitored refractive index changes
around an integrated microdisk-waveguide system by exciting the microdisk WGM and observing them through the
scattered light from the disk. In the present approach, we use
a dielectric sphere along with a single mode optical fiber as a
light input-output device.
Figure 1共a兲 depicts the geometric optics view of the
WGM resonance condition. In this view, the laser light
circles the interior of the sphere through multiples of total
internal reflections and returns in phase. The approximate
condition for optical resonance 共or WGM兲 is
a兲
Electronic mail: [email protected].
0021-8979/2010/107共10兲/103105/8/$30.00
共1兲
where, R and n1 are the sphere radius and refractive index, ␭
is the vacuum wavelength of light, and l is an integer. An
incremental change in the refractive index of the sphere will
result in a shift in the resonant wavelength; ⌬n1 / n1 ⬃ ⌬␭ / ␭.
Similarly, a change in the radius of sphere will also result in
a change in the resonant wavelength; ⌬R / R ⬃ ⌬␭ / ␭. Therefore, any change in the physical condition of the surrounding
that induces a ⌬n1 or ⌬R of the microsphere can be sensed
by monitoring the WGM shifts.
Equation 共1兲 is an approximation that helps us evaluate
the first order effects on WGMs of the sphere 共those due to a
a)
n1>n2
i
1
R
n1
2
…
N
 2  1
 2  1
…
1
Evanescent Wave
1
b)
i
i

FIG. 1. 共a兲 Ray optics model of light traveling inside sphere 共N is the
number of reflections for one round trip兲; 共b兲 reflection of a plane wave from
a dielectric interface.
107, 103105-1
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103105-2
J. Appl. Phys. 107, 103105 共2010兲
Ioppolo, Das, and Ötügen
change in R or n1兲. However, each time the light is reflected
off the inner surface of the sphere, the reflected wave experiences a certain “phase delay.” This phase delay is a function of the light wavelength, ␭, incidence angle, ␪i as well as
the sphere-to-surrounding refraction index ratio, n1 / n2.
Therefore, for a given laser wavelength and coupling angle,
the total optical path length for the round trip of the light is
somewhat different from 2␲Rn1 and is a function of the refractive index of the surrounding medium, n2. By tracking
the WGM shifts induced by perturbations in n2, the surrounding medium can be monitored provided that the induced shifts are in the order of ␦␭ or larger. However, n2
changes also affect the quality factor, Q and hence ␦␭. In the
following the effect of n2 on both WGM resonance and the
quality factor is analyzed using a simple geometric optics
and basic electromagnetic equations.
ZTE1 =
Z0
,
n1 cos ␪i
ZTE2 =
ZTM1 =
Z0 cos ␪i
,
n1
ZTM2 =
␪i ⬎ ␪c = sin−1
冑
␧2
n2
= sin−1 ,
␧1
n1
共2兲
where n and ␧ are the refractive indices and the dielectric
constants of the resonator 共1兲 and surrounding media 共2兲. The
reflection from such a planar interface should accurately
model each individual reflection in the ray-optics description
shown in Fig. 1共a兲. The reflection coefficients ⌫TE and ⌫TM
for the TE and TM polarizations are expressed using equivalent impedances of the two media as10
⌫TE =
where,
ZTE2 − ZTE1
,
ZTE2 + ZTE1
⌫TM =
ZTM2 − ZTM1
,
ZTM2 + ZTM1
共3兲
sin2 ␪i − n22
共4兲
,
− jZ0冑n21 sin2 ␪i − n22
n22
,
共5兲
with Z0 = 377 ⍀ as the wave impedance of free space. Our
interest is the phase associated with each reflection coefficient for large N.
Combining Eqs. 共3兲 and 共4兲 for the TE polarization,
⌫TE =
n1 cos ␪i + j冑n21 sin 2␪i − n22
n1 cos ␪i − j冑n21 sin 2␪i − n22
II. ANALYSIS
Starting with a quantum analogy-based theory developed
by Johnson7 describing the optical resonances of a sphere,
Teraoka et al.,8 carried out a perturbation approach to study
WGM shifts due to changes in the refractive index of the
medium surrounding the sphere. More recently, Schweiger
and Horn9 used a wave theory approach to obtain similar
results. We present here a simple approach using geometric
optics, along with basic electromagnetic wave reflection expressions, to obtain equations that accurately describe the
resonant wavenumber and quality factor Q of the WGM.
That is, using simple physical modeling of reflections at an
interface, we derive a modified version of Eq. 共1兲, together
with an expression for the associated Q, in which the effect
of n2 is included. The analysis is valid for R Ⰷ ␭ and the
sphere is treated as a cylinder for conceptual simplicity with
the light traveling on a plane parallel to the end surfaces of
the cylinder. When the order number l of resonance is large,
the cylindrical resonance condition would accurately model a
spherical resonance as well. The analysis yields two distinct
equations for the transverse electric 共TE兲 and transverse
magnetic 共TM兲 modes.
Consider the reflection of a plane wave from a dielectric
interface, as shown in Fig. 1共b兲, when the angle of incidence
␪i is larger than the critical angle. That is
− j冑
and
arg共⌫TE兲 = ␲ − 2 tan−1
A. WGM shift
Z0
n21
冑
共6兲
,
cos ␪i
冉冊
n2
sin ␪i −
n1
2
2
共7兲
.
Referring to Fig. 1共a兲, we note that, for N Ⰷ 1, the incidence
angle ␪i is
␪i =
␲ ␲ ␲
− ⬵ .
2 N
2
共8兲
Therefore, arg共⌫TE兲 can be approximated as follows:
arg共⌫TE兲 ⬵ ␲ −
2␲/N
冑 冉 冊
n2
1−
n1
2
共9兲
.
Similarly, combining Eqs. 共3兲 and 共5兲 for the TM polarization,
⌫TM =
− n22 cos ␪i − jn1冑n21 sin 2␪i − n22
+ n22 cos ␪i − jn1冑n21 sin 2␪i − n22
共10兲
,
yielding the argument
arg共⌫TM兲 = − 2 tan−1
cos ␪i
冉 冊冑
n1
n2
2
冉冊
n2
sin ␪i −
n1
2
2
.
共11兲
Again, Eq. 共11兲 can be approximated using Eq. 共8兲 as,
arg共⌫TM兲 ⬵ −
2␲/N
冉 冊冑 冉 冊
n1
n2
2
n2
1−
n1
2
.
共12兲
Equation 共9兲 shows that the reflection phase is close to ␲ for
the TE mode. We also know that the magnitude of reflection
is unity 共total internal reflection兲. Therefore, we may consider the case as a perturbation of an ideal conducting wall
for which 兩⌫兩 = 1 and arg共⌫兲 = ␲. Accordingly, the TE resonant condition for the dielectric sphere of Fig. 1共a兲 may also
be considered as a perturbation of that for a sphere with an
ideal conducting wall. Using multiple ray-reflection model
shown in Fig. 1共a兲, we consider the resonant condition for
the conducting sphere first. The resonance radius is analytically known from the first zero of the lth order Bessel
function10,11 that is k1R ⬵ l + 1.856l1/3. Here, k1 = 2␲ / ␭1 is the
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103105-3
J. Appl. Phys. 107, 103105 共2010兲
Ioppolo, Das, and Ötügen
wavenumber and ␭1 is the wavelength in the sphere. Thus,
from Fig. 1共a兲 and using Eq. 共8兲, the resonant condition can
be expressed as
N关2k1R cos ␪i − arg共⌫TM兲兴 = 2␲共l + 1.856l1/3兲,
and
␲
N2k1R cos ␪i ⬵ N2k1R = 2␲k1R = 2␲共l + 1.856l1/3兲.
N
k 1R ⬵
共13兲
The factor 1.856l1/3 in Eq. 共13兲 can be interpreted as a first
order correction that accounts for the spherical nature of the
wave front and the curvature of the resonator boundary,
which is different from the simple plane wave treatment of
Fig. 1共b兲. For large mode numbers 共l Ⰷ 1兲 this term would be
small compared to l, resulting in the approximate resonance
condition of Eq. 共1兲. The resonant condition for the dielectric
cavity of Fig. 1共a兲 may now be determined by including in
Eq. 共13兲 the phase contribution of Eq. 共9兲
2␲共l + 1.856l1/3兲 + arg共⌫TM兲N
␲
2N
N
⬵ l + 1.856l1/3 −
⌬共k1R兲 ⬵ −
共14兲
2␲共l + 1.856l1/3兲 + 关arg共⌫TE兲 − ␲兴N
␲
2N
N
冑
1
n2
1−
n1
冉冊
2.
=−
共15兲
Note that the final resonant condition does not depend on N,
which is the number of reflections the light ray experiences
for one full round trip in sphere as shown in Fig. 1共a兲. 共Also,
in principle, N need not be an integer.兲
The change in the resonant condition due to an incremental change in n2 can be evaluated by differentiating Eq.
共15兲.
⌬共k1R兲 ⬵ −
n2⌬n2
n21
冋 冉 冊册
n2
1−
n1
2 3/2
=−
n1n2⌬n2
,
关n21 − n22兴3/2
共16兲
or,
1
n2⌬n2
⌬共k1R兲 ⌬共k0R兲
=
=− 2
,
2 3/2
k 1R
k 0R
关n1 − n2兴 k0R
n2
n1
2
1
n2
1−
n1
冉冊
共19兲
2.
冦
冉冊
冋 冉 冊册 冋 冉 冊册
冋 冉 冊册
2n2⌬n2
n2⌬n2
2
n1
n21
+
n2 2 3/2
n2 2
1−
1−
n1
n1
n2
n1
2
n1n2⌬n2 2 −
冉冊
⬵ l + 1.856l1/3 −
冉冊
冉 冊冑
The above equation is the TM counterpart of Eq. 共15兲. Again,
the resonance shifts due to an incremental change in the refractive index of the surrounding medium can be obtained by
differentiating Eq. 共19兲
N兵2k1R cos ␪i − 关arg共⌫TE兲 − ␲兴其 = 2␲共l + 1.856l1/3兲,
k 1R ⬵
共18兲
n2
n1
⌬共k1R兲 ⌬共k0R兲
=
=−
k 1R
k 0R
,
冋 冉 冊册
n2⌬n2 2 −
关n21
冧
2
关n21 − n22兴3/2
or
1/2
−
n2
n1
n22兴3/2
共20兲
2
1
.
k 0R
共21兲
Equations 共17兲 and 共21兲 describe the dependence of the resonant frequency 共WGM兲 shift on the refractive index of the
medium surrounding the sphere for TE and TM polarizations, respectively. Note that l is kept constant in obtaining
the above derivatives. In an earlier study, Teraoka et al.8
carried out a perturbation analysis based on an equivalent
quantum theory model of the optical modes of spheres7 to
obtain expressions describing WGM dependence on ⌬n2. For
k0R Ⰷ 1, Eq. 共16兲 and the implicit formula obtained by Ref. 8
differs by the ratio of 冑兵关共l − 0.5兲 / 共k0R兲兴2 − n22其 / 共n21 − n22兲.
However, when the order number is approximated by l
= n1k0R their expression for WGM shift becomes identical to
Eq. 共16兲.
共17兲
where, k1 = k0n1.
The treatment for the TE polarization presented above
can be repeated for the TM polarization, noting only the key
differences. From Eq. 共12兲 we see that reflection phase for
TM polarization is close to zero for large N. Therefore, the
resonant dielectric sphere of Fig. 1共a兲 may be considered as a
perturbation of a sphere with an ideal magnetic wall for
which 兩⌫兩 = 1 and arg共⌫兲 = 0. As in the TE resonance case, the
TM resonant radius for the magnetic-walled resonator would
also correspond to the first zero of the lth order Bessel
function,10,11 with k1R ⬵ l + 1.856l1/3. So we have,
B. Quality factor
The resonance analysis presented above models the interface between the microsphere and the external medium
关Fig. 1共a兲兴 as a planar surface 关Fig. 1共b兲兴. This results in total
internal reflection, with magnitudes of the reflection coefficients equal to unity. However, each reflection generates
some radiation, attributed to the curvature of the interface.
We model the radiation loss with an equivalent radiation resistance Rre and Rrm, for TE and TM polarizations, respectively. The reflection coefficients for the TE and TM modes,
taking into account the equivalent radiation resistance are
given as
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103105-4
J. Appl. Phys. 107, 103105 共2010兲
Ioppolo, Das, and Ötügen
⌫TE =
ZTE2 + Rre − ZTE1
,
ZTE2 + Rre + ZTE1
⌫TM =
ZTM2 + Rrm − ZTM1
.
ZTM2 + Rrm + ZTM1
共22兲
The impedances ZTE1, ZTE2, ZTM1, and ZTM2 are given by
Eqs. 共4兲 and 共5兲. For N Ⰷ 1 and ␪i = ␲ / 2 − ␲ / N, the magnitudes of the reflection coefficients can be approximated as
兩⌫TE兩 ⬇ 1 −
2␲Rre
Z 1N
,
兩⌫TM兩 ⬇ 1 −
and
2RrmZ1␲
, 共23兲
N兩ZTM2兩2
where Z1 = 冑␮0 / ␧1 is the intrinsic impedance of the microsphere material. Equation 共23兲 indicates reflection coefficients smaller than unity, which is physically more realistic.
The complex resonance condition for the TE mode may now
be accurately established, using the reflection coefficient
magnitude of Eq. 共23兲
exp关− jN2k1R cos ␪i + jN arg共− ⌫TE兲兴兩⌫TE兩N
= f共␻r + j␻i兲 = exp共− jl2␲兲 = 1,
with
冉
f共␻r兲 = 兩⌫TE兩N = 1 −
2␲Rre
Z 1N
冊 冉
f共␻r + j␻i兲 = 1 ⬇ f共␻r兲 + j␻i
冉
⫻ 1−
冊
共24兲
N
⬇ 1−
冊
2␲Rre
,
Z1
冉
⳵ f共␻r兲
2␲Rk1␻i
= 1+
⳵␻
␻r
Prad = Pd2␲r2 =
共26兲
=
共27兲
The above equation shows that in order to evaluate the quality factor, the equivalent radiation resistance Rre has to be
calculated. With this objective, we consider a TE spherical
mode for r ⬎ R, with electric field distribution for ␪ = ␲ / 2
expresses as10
E␪ =
jl␾
Ah共2兲
l 共k2r兲e ,
共28兲
here A is a constant and h共2兲
l is the outgoing spherical Hankel
function of order l. When r → ⬁ the above equation may be
simplified to
E␪共r → ⬁兲 ⬇ Ajl+1
−jk2r
e
.
k 2r
2
兩A兩2兩h共2兲
l 共k2R兲兩
Rre2␲R2 .
兩ZTE2兩2
共32兲
Combining Eqs. 共31兲 and 共32兲, the equivalent radiation resistance can be expressed as
Rre =
兩ZTE2兩2
2
2
兩h共2兲
l 共k2R兲兩 Z2共k2R兲
共33兲
.
Using the above expression for Rre in Eq. 共27兲 the quality
factor Q can be derived as follows:
2
2
␻r Z 1Rk1 兩h共2兲
l 共k2R兲兩 Z2Z1共k2R兲 共Rk1兲
=
=
␻i
Rre
兩ZTE2兩2
⬇ y 2l 共k2R兲共k2R兲3
冋冉 冊 册
␧r1
−1 ,
␧r2
共34兲
where y l共 . 兲 is the spherical Bessel function of the second
kind. Note that we approximated 兩ZTE2兩 in the above derivation for ␪i ⬇ ␲ / 2 to obtain
兩ZTE2兩 =
where, ␻r and ␻i are real and imaginary angular frequencies.
Then, the quality factor, Q can be expressed as
␻r Z 1Rk1
=
.
Q=
␻i
Rre
兩E␪共R兲兩2
Rre2␲R2
兩ZTE2兩2
Prad = 兩H␾共R兲兩2Rre2␲R2 =
共25兲
2␲Rre
2␲Rk1␻i 2␲Rre
⬇1+
−
,
Z1
␻r
Z1
共31兲
The above Prad can also be written in terms of the electric or
magnetic field components at r = R, and the equivalent radiation resistance Rre.
Q=
冊
兩A兩22␲
.
Z2共k2兲2
␻␮0
冑k21 sin2 ␪i − k22 ⬇
Z2
冑冉 冊
k1
k2
=
2
−1
Z2
冑冉 冊
␧r1
−1
␧r2
共35兲
Using a similar procedure as for the TE mode, an expression
for the quality factor Q for TM mode can also be obtained.
exp关− jN2k1R cos ␪i + jN arg共− ⌫TM兲兴兩⌫TM兩N
= f共␻r + j␻i兲 = exp共− jl2␲兲 = 1,
冉
f共␻r兲 = 兩⌫TM兩N = 1 −
2␲Z1Rrm
兩ZTM2兩2N
冊 冉
N
⬇ 1−
共36兲
冊
2␲Z1Rrm
,
兩ZTM2兩2
共37兲
f共␻r + j␻i兲 = 1 ⬇ f共␻r兲 + j␻i
冉
共29兲
⫻ 1−
冉
⳵ f共␻r兲
2␲Rk1␻i
= 1+
⳵␻
␻r
冊
冊
2␲Rrm
2␲Rk1␻i 2␲Z1Rrm
⬇1+
−
,
兩ZTM2兩2
␻r
兩ZTM2兩2
The power density associated with the field is given by
兩E␪兩2
兩A兩2
=
,
Pd =
Z2
Z2共k2r兲2
Z2 = 冑␮0 / ␧2
.
共38兲
共30兲
is the intrinsic wave impedance of the
where
medium surrounding the sphere. The power radiated from
the sphere, per unit polar angle ⌬␪ = 1, can now be expressed
in terms of the power density.
Q=
␻r 兩ZTM2兩2Rk1
=
.
␻i
RrmZ1
共39兲
As in the TE mode case, we consider here a TM spherical
mode for r ⬎ R, with its magnetic field expressed for ␪
=␲/2
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103105-5
J. Appl. Phys. 107, 103105 共2010兲
Ioppolo, Das, and Ötügen
jl␾
H␪共r ⬎ R, ␪ = ␲/2兲 = Ah共2兲
l 共k2r兲e .
共40兲
The above equation may be simplified for r Ⰷ R, and the
corresponding power density Pd and radiated power Prad per
unit elevation angle can be expressed as
Pd = Z2兩H␪兩2 =
兩A兩2Z2
,
共k2r兲2
Prad = Pd2␲r2 .
l=1000
l=500
共41兲
Prad may also be expressed using the fields at r = R and the
equivalent resistance Rrm.
l=100
2
2
Prad = 兩H␪共R兲兩2Rrm2␲R2 = 兩A兩2兩h共2兲
l 共k2R兲兩 Rrm2␲R . 共42兲
Combining Eqs. 共41兲 and 共42兲, yields the equivalent resistance, Rrm, which can then be used in Eq. 共38兲 to find the
expression for the quality factor Q for the TM mode.
Rrm =
Q=
Z2
,
共2兲
兩hl 共k2R兲兩2共k2R兲2
共43兲
2
2
2
Rk1兩ZTM2兩2 兩h共2兲
l 共k2R兲兩 兩ZTM2兩 共k2R兲 共Rk1兲
=
RrmZ1
Z 1Z 2
⬇ y 2l 共k2R兲共k2R兲3
冉
␧r1
−1
␧r2
冊冉 冊
␧r1
.
␧r2
共44兲
Again, we approximated 兩ZTM2兩 in the above derivation for
␪i ⬇ ␲ / 2 to obtain:
兩ZTM2兩 =
冑k21 sin2 ␪i − k22
= Z2
␻␧2
冑冉 冊
␧r1
− 1.
␧r2
⬇ Z2
冑冉 冊
k1
k2
2
−1
共45兲
Fig. 2 shows the TE resonance quality factor as a function of
microsphere radius for a silica sphere surrounded by air
共n2 / n1 = 0.69兲. In the Q calculation, the order number, l, is
determined using 共a兲 the zeroth order approximation that includes only the reflection at the interface 关Eq. 共1兲兴; 共b兲 first
order approximation that includes reflection and surface curvature at the interface 关Eq. 共13兲兴; and 共c兲 second order approximation that includes reflection, surface curvature as
well as phase 关Eq. 共15兲兴. The resonance condition of Eq. 共1兲
significantly overestimates the quality factor. The conditions
of Eqs. 共13兲 and 共15兲 provide comparable values of the quality factor. However, the phase contribution in Eq. 共15兲 in-
FIG. 2. TE quality factor as a function of sphere radius. Order number l is
calculated using: 共a兲 Eq. 共1兲; 共b兲 Eq. 共13兲; and 共c兲 Eq. 共15兲.
FIG. 3. TE mode quality factor as a function of n2 / n1.
cludes the additional effect of the external medium, which is
valuable in the study of resonance perturbation due to variation in the external medium. The results using Eq. 共15兲 are
comparable to those from a much involved theory12 using a
volume-current analysis of the spherical mode. This illustrates the accuracy of the present method, obtained with analytical and conceptual simplicity. Figure 3 shows the effect
of refractive index ratio 共n2 / n1兲 and order number 共l兲 on TE
resonance quality factor. The quality factor, Q, increases with
increasing l and decreasing n2 / n1. For large l and small
n2 / n1 the Q value due to radiation is much larger than that
due to material loss. Accordingly, in this range the overall Q
is primarily determined by the material loss, and is essentially insensitive to the variations in the surrounding medium. On the other hand, for n2 / n1 ratios close to unity, the
contribution of radiation to Q value would be stronger making the overall quality factors smaller and more sensitive to
n2. This may adversely affect the accuracy of n2 measurements using WGM shifts. Yet, Q changes may be significant
enough to use them as a sensing parameter.
III. EXPERIMENTAL INVESTIGATION
A. Experimental setup
The output of the tunable distributed feedback laser diode was coupled into a single mode optical fiber with core
diameter of ⬃9 ␮m and cladding of 125 ␮m. The laser
beam is split in two by a beam splitter 共90%/10%兲 so that the
10% of the beam intensity goes to a photodiode to monitor
the laser beam intensity and use it as a reference signal. The
fiber with the 90% intensity was sent to the “sensor” side of
the fiber with the microsphere in contact with it. At its output
end, this fiber is connected to a fast photodiode to monitor
the transmission spectrum. This signal through the sensor
fiber was normalized using the reference signal. The microsphere was brought very close to the tapered fiber at a distance of the order of several nanometers to facilitate good
light coupling between fiber and resonator. The laser is tuned
by ramping the current into it using a laser controller. The
controller, in turn, was driven by a function generator that
provided a saw-tooth input to the controller. The output of
both photodiode was monitored by 16 bit digitizer 共data acquisition card兲 that was part of a host PC. The microspheres
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103105-6
J. Appl. Phys. 107, 103105 共2010兲
Ioppolo, Das, and Ötügen
Solution
a)
Microsphere
0.004
To temperature controller
 n2
From laser
Wolf et al (1965)
0.0035
0.003
 = 1312.59 nm
0.0025
 = 1312.46 nm
0.002
0.0015
0.001
0.0005
To photodiode
0
0
Distilled water
0.05
0.1
0.15
Molar concentration, g-mols/liter
FIG. 4. 共Color online兲 Details of the test section.
b)
0.004
were manufactured by melting a small section of an optical
fiber using a microtorch as was done in our previous
studies.2–5 The sphere diameters ranged between 350 and
500 ␮m in the present experiments.
0.0035
0.0025
 n2
The details of the test section are shown in Fig. 4. In
these experiments, the effect of potassium carbonate concentration 共in water兲 on the WGM shifts is studied. The microsphere is placed in deionized water in a container. A potassium carbonate solution was gradually added to the container
and allowed to diffuse for uniform concentration before the
WGM shifts are observed. The temperature of the solution
was monitored by a thermocouple and kept a constant value
of 22 ° C. The transmission spectrum was digitized and
stored on the host PC to determine the WGM shift. The
measurements were repeated with increasing concentrations
of the solution.
Using k = 2␲ / ␭, Eqs. 共18兲 and 共22兲 can be rewritten in
terms of wavelength, ␭ as;
⌬␭
␭
n2⌬n2
= 2
,
2 3/2
␭
共n1 − n2兲 2␲R
⌬␭
=
␭
冋 冉 冊册
n2⌬n2 2 −
共n21
−
n2
n1
n22兲3/2
 = 1312.59 nm
 = 1312.46 nm
0.002
0.0015
B. Experimental results
and
Wolf et al (1965)
0.003
共46兲
2
␭
,
2␲R
共47兲
for the TE and TM modes, respectively. Again, R and n1 are
the radius and index of refraction of the sphere, respectively.
共Note that the relationship between the WGM shifts and index of refraction change is not a function of the coupling
共launch兲 angle of the light into the sphere.兲
Figures 5共a兲 and 5共b兲 show the experimentally determined refractive index of potassium carbonate solution.
From the measured WGM shifts 共⌬␭兲, the changes in the
refractive index 共⌬n2兲 are determined using both the TE
mode using Eq. 共46兲 and shown in Fig. 5共a兲 and the TM
mode 关using Eq. 共47兲 and shown in Fig. 5共b兲兴. At this point,
we cannot determine if a mode observed in the transmission
spectrum is TE or TM. As one would expect, the trends given
by the two modes are essentially the same. In Fig. 5, refractive index dependence on potassium carbonate concentration
0.001
0.0005
0
0
0.05
0.1
0.15
Molar concentration g-mols/liter
FIG. 5. 共Color online兲 共a兲 Dependence of n2 on potassium carbonate concentration 共TE mode used for experimental results兲; 共b兲 dependence of n2 on
potassium carbonate concentration 共TM mode used for experimental
results兲.
from the chemical literature13 is also presented for comparison. 共Note that the data in Ref. 13 are for ␭ ⬃ 590 nm.兲 The
agreement between the experiments and the literature data is
stronger in Fig. 5共a兲 than that in Fig. 5共b兲. This seems to
indicate that in all the experiments, the observed WGMs happen to be TE modes. However, in the results summarized by
this figure, we cannot independently identify the observed
WGMs as either TE or TM modes. In order to clarify this
point, an additional experiment was carried out where ⌬␭
shifts were measured while simultaneously identifying the
observed WGMs as TE or TM. To accomplish this, in addition to the transmission spectrum through the fiber, scattered
light from the microsphere was also monitored. This made it
possible to choose the appropriate equation 关Eq. 共46兲 or 共47兲兴
to determine ⌬n2. The experimental arrangement is shown in
Fig. 6. At resonance, the constructive build-up of the electromagnetic field in the sphere leads to increased scattering 共radiation兲 through the sphere surface 共corresponding to the
WGM dips in the transmission spectrum兲. The scattered light
from the sphere is filtered spectrally 共using a laser line filter兲
and spatially 共using a pinhole兲 and then imaged on a photomultiplier tube using a pair of lenses. A linear polarizer that
is placed between the sphere and the front collecting lens
allows for the identification of the observed mode as TE or
TM. Note that the polarization of a mode in the sphere is
predominantly in the direction normal to the plane in which
the light circulates 关as defined in Fig. 1共a兲兴.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
103105-7
J. Appl. Phys. 107, 103105 共2010兲
Ioppolo, Das, and Ötügen
½ wave
plate
solution
Solution
From laser
Photo diode
Linear polarizer
DAQ
Collecting lenses
PC
PMT
FIG. 6. 共Color online兲 Setup for index of refraction measurements with
simultaneous TE and TM detection.
The container in Fig. 6 is initially filled with an aqueous
solution of methanol 共CH3OH兲. Pure water is then gradually
added and from the observed WGM shifts 共⌬␭兲 the refractive
index, n2, is determined using either Eq. 共46兲 or 共47兲, as
appropriate. Figures 7共a兲 and 7共b兲 show typical scattering
spectra for two mixture-concentrations. Figure 7共a兲 is for the
case when the polarizer is adjusted such that only the TM
modes are observed. In Fig. 7共b兲, the polarizer is adjusted
such that the TE spectra is observed. In both figures, the
simultaneously observed transmission spectra are also plot-
ted. In these figures, C1 and C2 refer to two different concentrations of methanol solutions 共C2 representing a higher
concentration of water兲.
Figure 8 shows the experimentally determined refractive
index of different concentrations of aqueous solution of
methanol along with those values from the literature. Experimental data is obtained by the sensor using both the TE and
TM modes. The figure shows that when the appropriate
mode equation is used 关i.e., Eq. 共46兲 for the TE modes and
Eq. 共47兲 for the TM modes兴, the agreement between the published literature and the present results is strong. This result
shows the potential of the proposed microsphere concentration sensor. However, additional work is needed to develop
robust methods either to identify TE and TM modes in the
transmission spectra or to selectively excite the sphere with
TE or TM modes.
IV. CONCLUSIONS
The simple ray tracing, plane wave approach used in the
present analysis results in equations for ⌬␭ dependence on
⌬n2 that are identical to those approximate explicit expressions previously derived from a perturbation analysis based
on a quantum-analogy model. However, we derive the explicit Eqs. 共15兲 and 共19兲 for the resonant wavelength using
simple reflection principles, from which the ⌬␭ dependence
0.8
a)
0.7
3.5
3
0.5
Transmitted C1
Transmitted C2
Scattered C2
Scattered C1
0.4
2.5
0.3
Scattered light (au)
Transmitted
light (au)
Trasmitted light (au)
0.6
0.2
2
0.1
1.5
1.31278
0
1.31279
1.3128
1.31281
1.31282
1.31283
1.31284
1.31285
l, mm
m
1
b)
0.9
9
0.8
Trasmitted light (au)
Transmitted light (au)
8
0.7
Transmitted C1
Transmitted C2
7
0.6
Scattered C1
6
Scattered C1
0.4
5
0.3
4
0.2
3
2
1.31286
0.5
Scattered light (au)
10
0.1
1.31286
1.31287
1.31287
1.31288
1.31288
1.31289
1.31289
1.31290
1.31290
0
1.31291
l, mm
FIG. 7. 共Color online兲 Typical scattering spectrum along with the corresponding transmission spectrum; 共a兲 TM mode and 共b兲 TE mode.
Author complimentary copy. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp
103105-8
1.344
1.343
n2
J. Appl. Phys. 107, 103105 共2010兲
Ioppolo, Das, and Ötügen
Wolf et al (1965)
1.342
Experiment, TM mode
1.341
Experiment, TE mode
1.34
1.339
1.338
1.337
low atmospheric pressures. However, this also means that in
other sensor applications 共such as temperature, pressure, or
force兲 gaseous contaminants in air would not have a significant effect on signal. For a typical liquid application Eqs.
共46兲 and 共47兲 are distinct enough that the nature 共TE or TM兲
of the WGM tracked needs to be known so that the appropriate equation is used.
1.336
1.335
15.000
17.000
19.000
21.000
Molar concentration of AM g-mols/liter
23.000
FIG. 8. 共Color online兲 Dependence of refractive index on molar concentration of aqueous methanol solution.
ACKNOWLEDGMENTS
This research effort was supported by the National Science Foundation through Grant No. CBET-0502421.
G. Guan, S. Arnold, and M. V. Ötügen, AIAA J. 44, 2385 共2006兲.
T. Ioppolo, M. Kozhevnikov, V. Stepaniuk, M. V. Ötügen, and V. Sheverev, Appl. Opt. 47, 3009 共2008兲.
3
T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, J. Appl. Phys. 105, 013535
共2009兲.
4
T. Ioppolo and M. V. Ötügen, J. Opt. Soc. Am. B 24, 2721 共2007兲.
5
T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, Opt. Express 17, 16465 共2009兲.
6
E. Krioukov, D. J. W. Klunder, A. Driessen, J. Greve, and C. Otto, Opt.
Lett. 27, 512 共2002兲.
7
B. R. Johnson, J. Opt. Soc. Am. A 10, 343 共1993兲.
8
I. Teraoka, S. Arnold, and F. Vollmer, J. Opt. Soc. Am. B 20, 1937 共2003兲.
9
G. Schweiger and M. Horn, J. Opt. Soc. Am. B 23, 212 共2006兲.
10
R. F. Harrington, Time-Harmonic Electromagnetic Fields 共Wiley
Interscience-IEEE, New York, 2001兲.
11
Handbook of Mathematical Functions, edited by M. Abramowitz and I. A.
Stegun 共Dover, New York, 1970兲.
12
B. E. Little, J.-P. Laine, and H. A. Haus, J. Lightwave Technol. 17, 704
共1999兲.
13
A. V. Wolf, G. Morden, and M. G. Brown, Handbook of Chemistry and
Physics, 46th ed. edited by R. C. Weast and S. M. Selby 共CRC Press, Boca
Raton, FL, 1965兲, p. D-129.
1
on ⌬n2 can be obtained by differentiation. Equations 共15兲
and 共19兲 are essentially more accurate form of Eq. 共1兲, that
include additional terms. The second term is identical for
both the TE and TM modes, and represents the effect of the
curvature of the sphere surface. For large l, this term becomes negligible compared to the first term. The last term is
the effect of refractive index of the surrounding medium on
the resonance condition. For large l, this term also is dominated by the first term on the right hand side. The quality
factor analysis shows that the energy losses due to surface
radiation are very small 共the Q-factors are generally large
even for small order numbers兲. Equations 共46兲 and 共47兲 describe the dependence of WGM shifts on the refractive index
change in the medium surrounding the sphere. The equations
suggest that the proposed concentration sensor would not be
sensitive enough for applications in a gas medium at or be-
2
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