Download Theory of resonance shifts in TE and TM whispering

I. Teraoka and S. Arnold
Vol. 23, No. 7 / July 2006 / J. Opt. Soc. Am. B
1381
Theory of resonance shifts in TE and TM
whispering gallery modes by nonradial
perturbations for sensing applications
Iwao Teraoka and Stephen Arnold
Microparticle Photophysics Laboratory, Polytechnic University, Brooklyn, New York 11201
Received September 21, 2005; revised February 13, 2006; accepted February 17, 2006; posted March 7, 2006 (Doc. ID 64876)
A perturbation theory is presented for the frequency shift of highly resonant photonic whispering gallery
modes in a transparent sphere. Using a vector wave equation, we derive a general formula for the shifts in TE
and TM polarization by adsorption of another dielectric medium. The adsorbed medium can have an arbitrary
shape and refractive-index profile. The formula is applied to adsorption of a thin layer and deposition of a small
spherical particle, many such particles, and thin cylindrical particles on the resonator surface. We found that
the ratio of the TM mode shift to the TE mode shift is sensitive to the shape of the adsorbates and their orientation. Calculation results are discussed in terms of a dipolar field. © 2006 Optical Society of America
OCIS codes: 170.4520, 300.6490, 260.2110.
ratio of the TE and TM modes can provide information on
the shape and orientation of adsorbed particles.
1. INTRODUCTION
Resonant photonic modes called whispering gallery
modes (WGMs) in a transparent sphere have high sensitivity to environmental changes.1–7 The sensitivity derives from the ability of the evanescent field of a WGM to
explore the surrounding medium. The change is most sensitively detected in the frequency of the extremely narrow
resonance mode.8,9
Earlier, we developed a perturbation theory for sensing
application of the frequency shift. First3 we considered
adsorption of a small dielectric medium onto the spheres’s
surface.3 The coupling of the evanescent field with the adsorbate causes the mode energy of the WGM to change.
By equating the ratio of the frequency shift to the unperturbed frequency and the ratio of the energy change to the
unperturbed mode energy, we derived a general formula
for the shift of the TE mode resonance. Second,10 we applied a rigorous perturbation theory to the wave equation
to derive a formula for the shifts of TE and TM modes
caused by small refractive-index changes in the
surrounding.10 The changes were limited to those given
by a radial function.
In the present paper we describe a general theory for
the resonance frequency shift of a WGM that is due to a
change in refractive index in a region of arbitrary geometry near or on the spherical resonator. The change in refractive index does not need to be small. We use the vector
wave equation to derive the general formula of the shift,
an approach similar to the one adopted by Harrington to
estimate the shift of a cavity mode.11 We then apply the
general formula to adsorption of a thin layer onto the
resonator and the deposition of a small spherical particle
and a cylindrical particle. We demonstrate that the shift
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2. PERTURBATION THEORY
A. General Theory
Consider a transparent sphere of volume V1 and uniform
refractive index n1 suspended in a medium of uniform refractive index n2 (n1 ⬎ n2; both are real). The relative electric permittivity ⑀r共r兲 at r in this unperturbed system is
given as
⑀r共r兲 =
再
n 12 , r 苸 V 1
冎
n22 , otherwise
共1兲
.
When a dielectric medium of volume Vp and refractive index np共r兲 is brought near the sphere’s surface, as shown
in Fig. 1, the change in the relative permittivity is given
as
␦⑀r共r兲 =
再
n p2 − n 22 , r 苸 V p
0,
冎
otherwise
.
共2兲
For simplicity, we assume that ␦⑀r共r兲 changes smoothly
with r within Vp. We allow ␦⑀r共r兲 to be locally large. A
typical example is to bring a globular protein molecule
close to the surface of a silica sphere 100 ␮m in diameter
suspended in water.
The change in refractive index in Vp causes the resonance wave vector of the WGM to change from k0 to k and
the spatial part of the electric field to change from E0共r兲
to E共r兲. Unperturbed field E0 satisfies the vector wave
equation:
© 2006 Optical Society of America
1382
J. Opt. Soc. Am. B / Vol. 23, No. 7 / July 2006
I. Teraoka and S. Arnold
Fig. 1. (Color online) Left, particle Vp placed near the surface of
a transparent microsphere V1. The rest of the microsphere is V2.
A fiber is side coupled to the sphere. Right, cross-sectional view of
the electric field (red online) and the particle, shown for the firstorder mode.
ⵜ ⫻ ⵜ ⫻ E 0 = k 02⑀ rE 0 .
共3兲
The time dependence of E0 is described by exp共−i␻0t兲,
where ␻0 = ck0 and c is the velocity of light in vacuum.
Perturbed field E follows
ⵜ ⫻ ⵜ ⫻ E = k2共⑀r + ␦⑀r兲E.
共4兲
*
We multiply E0 , the complex conjugate of E0, from left of
Eq. (4) and integrate the product over the entire space V:
冕
*
E0 · ⵜ ⫻ ⵜ ⫻ Edr = k2
V
冕
冕
E0 =
E0 · ⑀rEdr
+k
2
E0 · ␦⑀rEdr.
共5兲
Appendix A shows that the integral of E0* · ⵜ ⫻ ⵜ ⫻ E is
identical to the integral of E · ⵜ ⫻ ⵜ ⫻ E0*. With the complex conjugate of Eq. (3), we have
E0 · ⵜ ⫻ ⵜ ⫻ Edr =
V
冕
*
E · ⵜ ⫻ ⵜ ⫻ E0 dr
V
= k 02
冕
E · ⑀rE0*dr.
共6兲
V
From Eqs. (5) and (6) we have
共k02 − k2兲
冕
E · ⑀rE0*dr = k2
V
冕
E0* · ␦⑀rEdr.
共7兲
V0
Equation (7) applies to both TE and TM modes. We let k
= k0 + ␦k and consider first-order perturbation. We obtain
the fractional shift as
␦k
k0
=−
冕
冕
␦⑀rE0* · Epdr
Vp
2
exp共im␸兲
k 0r
where S0共r兲 satisfies
*
Vp
冕
B. TE Mode Energy
This subsection and Subsection 2.C will review a theory
for a WGM in a dielectric sphere.12 They are necessary for
our application of the general theory in Section 3.
We set up spherical polar coordinates 共r , ␪ , ␸兲 with the
origin at the center of the resonator sphere. For the TE
mode specified by polar index l and azimuthal index m
共m = −l , −l + 1 , . . . , l兲, E0 is given as12
*
V
*
surface. It is possible to apply his theory by placing our
resonator systems within a large cavity. As in our derivation, his formula applies to any mode in a cavity. Hence, it
is not necessary to adjust the size and shape of the enclosing cavity such that the frequency of our WGM matches
one of the cavity modes. The resonator and the surrounding medium alone within the cavity cause a frequency
shift. With the particle of np added, the shift is slightly
different. The difference in the frequency shift is what we
obtained as ␦k in Eq. (8).
,
共8兲
⑀rE0 · E0dr
*
V
where E in the denominator was replaced with E0. The
electric field may change exterior to Vp, but it does not
contribute to ␦k in the first-order perturbation. The shift
is equal to the perturbation integral divided by the mode
energy integral.
When ␦⑀r is small everywhere and only then, Ep in Eq.
(8) is replaced with E0 in the first-order perturbation.
Then ␦⑀rE · E0 = ␦⑀r兩E0兩2, and we recover the formula obtained earlier.10
Harrington11 derived a similar formula for an electromagnetic wave within a cavity that has a metallic interior
冋
⳵2
⳵r
l共l + 1兲
2
−
r2
册
S0共r兲Xlm共␪兲,
共9兲
S0 = − ⑀r共r兲k02S0 .
共10兲
The angular vector function Xlm共␪兲 is given as
Xlm共␪兲 =
im
sin ␪
Plm共cos ␪兲ê␪ −
⳵
⳵␪
Plm共cos ␪兲ê␸ ,
共11兲
where Plm共x兲 is the associated Legendre function and ê␪
and ê␸ are unit vectors in the relevant directions. In this
coordinate system, n共r兲 = n1 for r ⬍ a and n2 for r ⬎ a,
where a is the resonator radius.
For a steady-state WGM with real k0 at resonance,
S0共r兲 is expressed as12
S0共r兲 =
再
␺l共n1k0r兲,
r⬍a
Bl␹l共n2k0r兲, r ⬎ a
冎
,
共12兲
where the coefficient for r ⬍ a is set to 1 and Bl is determined from the continuity: ␺l共n1k0a兲 = Bl␹l共n2k0a兲. Here,
␺l共z兲 = zj / 共z兲 and ␹l共z兲 = zn / 共z兲 are spherical Ricatti–Bessel
and Ricatti–Neumann functions, where jl共z兲 and nl共z兲 are
spherical Bessel and Neumann functions, respectively.
The resonance condition
n1
␺l⬘共n1k0a兲
␺l共n1k0a兲
= n2
␹l⬘共n2k0a兲
␹l共n2k0a兲
共13兲
gives k0, where the prime denotes the derivative of a function by its argument. The exterior solution at resonance
decays nearly exponentially.
For a given l, there can be more than one value of k0a
that satisfies Eq. (13). These resonance modes are called
the first-order mode 共v = 1兲, the second-order mode 共v = 2兲,
etc. in increasing order of k0a. As Eq. (13) indicates, the
resonance condition is independent of m for a perfectly
spherical resonator. The 2l + 1-fold degeneracy is lifted by
any symmetry-breaking perturbation such as a spheroidal shape of the resonator.13
I. Teraoka and S. Arnold
Vol. 23, No. 7 / July 2006 / J. Opt. Soc. Am. B
Let Wlm be the integral of 兩Xlm共␪兲兩2 over the entire solid
angle:
Wlm ⬅
冕
␲
sin ␪d␪
0
冕
2␲
d␸兩Xlm兩2 =
0
4␲共l + m兲!l共l + 1兲
共l − m兲!共2l + 1兲
冕
⑀rE0* · E0dr = Wlm
.
a
冕
⑀rE0 · E0dr =
*
Wlm
k 02
冕
⬁
⑀r共r兲关S0共r兲兴2dr.
共15兲
0
The integral in Eq. (15) does not exactly converge for
S0共r兲 given by Eq. (12) because of a sinusoidal component
in S0共r兲 at large r. However, use of a convergence coordinate forces Eq. (15) to reduce to10
冕
⑀rE0* · E0dr = Wlm
a
2k02
共n12 − n22兲关S0共a兲兴2 .
共16兲
Another way to force the convergence was proposed by
Lai et al. for an exterior solution that becomes an outgoing wave at large r (quasi-normal mode).13
C. TM Mode Energy
For the TM mode,
components12:
E0 =
冋
exp共im␸兲 1
2 2
k0 n 共r兲
r
has
E0
angular
T0⬘共r兲Ylm共␪兲 +
1
r2
and
radial
册
共18a兲
Zlm共␪兲 = l共l + 1兲Plm共cos ␪兲êr ,
共18b兲
where êr is the unit vector in the radial direction.
The solution T0共r兲 of a steady-state TM mode is essentially identical to Eq. (12) with a different coefficient Al
for r ⬎ a. The resonance condition is slightly different12:
1 ␺l⬘共n1k0a兲
n1 ␺l共n1k0a兲
=
1 ␹l⬘共n2k0a兲
n2 ␹l共n2k0a兲
.
共19兲
The integral of 兩Ylm共␪兲兩2 over the solid angle is the same
as that of 兩Xlm共␪兲兩2. The integral of 兩Zlm共␪兲兩2 is l共l + 1兲 times
as large. Therefore the mode energy integral is
冕
⑀rE0* · E0dr =
Wlm
k 04
冕
⬁
0
1
n4共r兲
l共l + 1兲
+
r2
再
冎
关T0共r兲兴2 dr.
冕
␦⑀rE0* · Epdr =
Wlm
k 02
冕
冎
.
共21兲
a+t
␦共n2兲关S0共r兲兴2dr.
共22兲
a
Since t Ⰶ ␭, S0共r兲 in Eq. (22) can be replaced by S0共a兲.
From Eqs. (8), (16), and (22), we have the TE shift as
共␦k/k0兲TE = −
1
a共n12 − n22兲
冕
a+t
␦共n2兲dr.
共23兲
a
For the TM mode, E0 and Ep have both angular and radial components in the perturbation integral in Eq. (8).
The boundary condition requires that Ep be treated explicitly:
Ep =
冋
exp共im␸兲 1
2
k0 np
2
r
T⬘共r兲Ylm共␪兲 +
1
r2
册
T共r兲Zlm共␪兲 ,
共24兲
where T共r兲 and its derivative T⬘共r兲 account for the field
within the layer. They are, in general, different from T0共r兲
and T0⬘共r兲, the unperturbed functions. Angular integration of ␦⑀rE0* · Ep leads to
冕
␦⑀rE0* · Epdr = −
Wlm
k 04
冕
a+t
a
l共l + 1兲
Calculation of Eq. (20) suffers from the same problem as
that for the TE mode. However, using the same method as
the one used for the TE mode yields10
共n1k0a兲2
A. Thin-Layer Adsorption
The first example is adsorption of a thin layer of uniform
thickness t (t Ⰶ ␭ = 2␲ / k0; t Ⰶ a) and refractive-index profile np共r兲. Vp is specified by a ⬍ r ⬍ a + t. We allow ␦⑀r
= ␦共n2兲 = 关np共r兲兴2 − n22 to change as a function of r within
the layer.
For the TE mode, both E0 and Ep in the perturbation
integral in Eq. (8) have only angular components. Their
continuity across the two interfaces at r = a and a + t allows Ep to be replaced with E2, which is further replaced
with E02 in the first-order perturbation. Therefore, with
Eqs. (9) and (14), angular integration of ␦⑀rE0* · Ep leads
to
+
共20兲
l共l + 1兲
+
3. EXAMPLES OF TE AND TM SHIFTS
Vp
关T0⬘共r兲兴2
␹l共n2k0a兲
2
The mode energy can be divided into tangential and radial components. A typical value of their ratio,
共n1k0a兲2共␹l⬘ / ␹l兲2 / l共l + 1兲, is ⬃0.163, where n1 = 1.47, n2
= 1.33, l = 500, and v = 1 共k0a = 349.15兲 were used. The radial component dominates the energy of a TM mode.
Vp
Ylm共␪兲 = êr ⫻ Xlm共␪兲,
␹l⬘共n2k0a兲
⫻关T0共a兲兴2
T0共r兲Zlm共␪兲 . 共17兲
Scalar function T0共r兲 satisfies the same equation as the
equation that is satisfied by S0共r兲 in the interior and the
exterior of the resonator. The angular functions Ylm共␪兲
and Zlm共␪兲 are given, respectively, as
−1
2k02 n22
共14兲
Then the mode energy integral in the denominator of Eq.
(8) is calculated as
冉 冊
再冋 册
n 12
1383
r2
冋
册
␦共n−2兲 T⬘共r兲T0⬘共r兲
T共r兲T0共r兲 dr,
共25兲
where ␦共n−2兲 = 关np共r兲兴−2 − n2−2. Since t Ⰶ ␭, T0共r兲 and T0⬘共r兲
can be replaced by T0共a兲 and T0⬘共a+兲, respectively, where
a+ is infinitesimally greater than a. Likewise, T共r兲 and
T⬘共r兲 can be replaced by T共a兲 and T⬘共a+兲. Since the change
of T共r兲 from T0共r兲 at r ⬍ a is only the shift in the wave vec-
1384
J. Opt. Soc. Am. B / Vol. 23, No. 7 / July 2006
I. Teraoka and S. Arnold
tor, T共r兲 = T0共r兲 for r ⬍ a in the first-order perturbation.
The boundary condition gives T共a兲 = T0共a兲 and
T⬘共a+兲 / np12 = T0⬘共a+兲 / n22, where np1 = np共a+兲.
With T0共a兲 = Al␹l共n2k0a兲 and T0⬘共a+兲 = Aln2k0␹l⬘共n2k0a兲,
Eq. (25) leads to
冕
␦⑀rE0* · Epdr = −
Vp
Wlm
k 02
共k0a兲2
冕
a
册冕
a+t
␦共n−2兲dr,
共26兲
a
n22np12
2
2
a共n1 − n2 兲
冕
a+t
␦共n−2兲dr,
␦共n2兲dr
As we will see below, the ratio is equal to Rl when ␦n2 is
uniform.
The solid, dashed, and the dashed–dotted curves in Fig.
2 represent plots of R250 as a function of np1 for v
= 1 , 2 , 3, respectively. The parameters used are n1 = 1.47
and n2 = 1.33. The resonance wave vectors (TM) for the
three mode orders are k0a ⬵ 176.70, 182.62, 187.75, respectively. The approximation by Eq. (29) with np = np1 is
shown for v = 1 by a dotted curve that nearly overlaps the
solid curve. The approximation is equally good for v = 2
and v = 3 (not shown) when l is as large as the one used
here.
Equations (23), (27), and (30) apply to arbitrary
refractive-index profiles in the adsorption layer. For a
uniform refractive index, np2共r兲 = np2,
冉 冊
␦k
k0
=−
TE
n p2 − n 22 t
n 12 − n 22 a
,
共31兲
a
共␦k/k0兲TM
共␦k/k0兲TE
where Rl is defined as
l共l + 1兲共np−2 − n1−2兲x−2
共␹l⬘/␹l兲2 + l共l + 1兲共n1x兲
.
−2
共28兲
When ␦共n2兲 is small, Eqs. (23) and (27) are identical to the
results obtained in the first-order perturbation of wave
equations for small refractive-index changes.10 For given
n1, n2, and np, Rl depends on l and k0a. For large l or
large k0a, which is the case for WGM sensors, ␹l⬘ / ␹l
⬵ −关l2 / 共n2k0a兲2 − 1兴1/2. Then,
Rl共x;n1,n2,np兲 ⬅ 1 +
Rl共k0a;n1,n2,np1兲.
a+t
共30兲
共27兲
Rl共x;n1,n2,np兲 ⬅ 1 +
=−
␦共n−2兲dr
a
where ␹l and ␹l⬘ are evaluated at z = n2k0a. This perturbation integral is dominated by the contribution of the radial component of electric field, as in the mode energy integral. The ratio of the intensity of the tangential
component to that of the radial component is
共np1k0a兲2共␹l⬘ / ␹l兲2 / l共l + 1兲, which is greater than the counterpart in the mode energy integral when np1 ⬎ n1.
Using Eqs. (8), (21), and (26), we obtain the fractional
shift for the TM mode, 共␦k / k0兲TM, which can be separated
into the factor determined by the refractive-index profile
and the rest by Rl:
共␦k/k0兲TM = Rl共k0a;n1,n2,np1兲
共␦k/k0兲TE
冕
a+t
关T0共a兲兴2 np12共␹l⬘/␹l兲2
l共l + 1兲
+
冋
共␦k/k0兲TM
n22np12
np−2 − n1−2
n2−2 − 共k0a/l兲2 + n1−2
.
= Rl共k0a;n1,n2,np兲.
共32兲
For n2 = 1 (air), these results reduce to those derived by
Folan.15 The TM-to-TE shift ratio is equal to Rl that is
shown in Fig. 2. When np ⬎ n1, the TM mode shifts less
than the TE mode does, because the boundary condition
at the sphere–layer interface causes a smaller amplitude
of Ep than does E1, whereas Ep = E1 in the TE mode. The
共29兲
The dependence of Rl on k0a and l is mostly through the
ratio of k0a to l. The resonance wave vector is related to l
and mode order v 共v = 1 , 2 , . . . 兲 by k0a / l ⬵ n1−1共1 + ␤vl−2/3兲,
where ␤v = 2−1/3␣v, where ␣v is the vth root of the Airy
function Ai共−z兲; ␤1 ⬵ 1.856, ␤2 ⬵ 3.245, ␤3 ⬵ 4.382, . . ..14
Since l−2/3 Ⰶ 1, we find that Rl at resonance depends only
weakly on l and v and is determined primarily by n1, n2,
and np. The above relationship further simplifies Eq. (29).
Equation (23) indicates that ␦k / k0 of the TE mode is
determined solely by the refractive-index profile of the adsorption layer and is common to the WGMs of different
values of l, m, and v. The fractional shift does not depend
on the wavelength of the laser used in the measurement
either. The fractional shift of the TM mode is also determined primarily by the refractive-index profile and depends only weakly on l and v.
Comparison of Eqs. (23) and (27) gives the TM-to-TE
shift ratio as
Fig. 2. (Color online) Factor Rl plotted as a function of refractive
index np1 in the immediate neighborhood of the resonator’s surface in the adsorption layer. The parameters used are n1 = 1.47,
n2 = 1.33, and l = 250. The solid curve (red online) is for the firstorder mode 共v = 1兲, and the dotted curve, nearly overlapping the
solid curve, represents the approximation formula by Eq. (29).
The factors for v = 2 , 3 are shown as a dashed and a dashed–
dotted curve, respectively.
I. Teraoka and S. Arnold
Vol. 23, No. 7 / July 2006 / J. Opt. Soc. Am. B
shift ratio will be useful for estimating np when the layer
is uniform and sufficiently thin 共t Ⰶ ␭兲. There is no need to
know t.
For given n1, n2, and np, 共␦k / k0兲TM ⬀ Rl. From Fig. 2 we
find that 兩共␦k / k0兲TM兩 for v = 2 and v = 3 at np = 1.6 are only
1.2% and 1.7%, respectively, less than that for v = 1. If the
TM resonance shift is used in the WGM sensor to estimate the thickness of adsorption layer, the error that is
due to an unidentified mode order is at most 1.7% for l
= 250. Use of a spherical resonator of a larger diameter or
a shorter wavelength of light or both increases l, thus decreasing the error associated with the unidentified mode
order.
We note that the fractional shift of the TE mode is insensitive to internal rearrangement of the adsorption
layer because 兰 ␦共n2兲dr is held unchanged. The shift
changes for the TM mode, however.
B. Adsorption of a Single Particle
In this subsection we examine the TE and TM shifts by
adsorption of a single spherical particle onto the resonator’s surface. Such shifts will be compared with the shifts
caused by many such particles to form a packed monolayer. We also consider TE and TM shifts by a single cylindrical particle. We assume, for simplicity, that each of
these particles has a uniform refractive index and that its
linear dimension is sufficiently small compared with the
wavelength of light that the field within the particle can
be regarded as static and uniform.11
A small spherical particle (diameter 2b Ⰶ ␭) with uniform refractive index np is adsorbed onto the resonator at
共a , ␪ , ␸兲. If the particle were placed alone in a medium of
refractive index n2 and a uniform field E02, field Ep within
the particle would be uniform and given as16
Ep =
3n22
2n22 + np2
共33兲
E02 .
When it is placed on the resonator’s surface, the polarized
particle will create additional polarization in the resonator, which in turn causes additional polarization within
the particle. This effect does not show in the integral of
␦⑀rE0* · Ep in Eq. (8) over the volume of the particle in any
polarization directions, as proved in Appendix B. Therefore we can use Ep above for the perturbation integral in
both TE and TM modes. In the small volume Vp, 兩E0兩2 is
constant. Therefore
冕
␦⑀rE0 · Epdr = 共np − n2 兲
*
2
Vp
2
3n22
2n22 + np
冕
共␦k/k0兲TE = − 共␩/Wlm兲共Vp/a3兲兩Xlm共␪兲兩2 ,
␩⬅
3n22
关S0共a兲兴2
2n22 + np2 k02a2
n p2 − n 22
Vp兩Xlm共␪兲兩2 .
Then, from Eqs. (8), (16), and (35), we obtain
3n22
n12 − n22 2n22 + np2
共37兲
.
The fractional shift is identical for m and −m.
The fractional shift of the TE mode is proportional to
the volume ratio of the adsorbed particle to the resonator.
The sensitivity of the frequency shift depends on 兩Xlm兩2,
the squared magnitude of the electric field on the spot
where the particle is adsorbed. Figure 3 shows
兩Xlm兩2 / Wlm = −共a3 / Vp兲共␦k / k0兲TE / ␩ as a function of ␪ for
l = 500, v = 1, and five values of 兩m兩. All of the five curves
have different traces. This means that the adsorption of a
small particle will lift the 2l + 1-fold degeneracy of a WGM
into l + 1 resonance lines. The mode with 兩m兩 = l is sensitive
to the adsorption near the equator. Maximizing the sensitivity of the WGM adsorption sensor is achieved by directing the analyte particles to the equator or by immobilizing the binding sites at the equator.5 The mode with 兩m兩
= l − 1 has two bands of sensitive region. With the spot of
adsorption further away from the equator, the most sensitive mode changes to one with a smaller 兩m兩.
For the TM mode, use of Eq. (17) for E0 allows Eq. (34)
to be rewritten as
冕
␦⑀rE0 · Epdr = 共np − n2 兲
*
2
Vp
⫻
2
冋冉 冊
␹ l⬘
␹l
3n22
关T0共a兲兴2
2n22 + np2 共n2k0a兲2
2
兩Ylm共␪兲兩2 +
Vp
兩Zlm共␪兲兩2
共n2k0a兲2
册
,
共38兲
where ␹l and ␹l⬘ are evaluated at z = n2k0a. With Eqs. (8)
and (21), we obtain
共␦k/k0兲TM = −
Vp
共36兲
where
V 兩E 兩 , 共34兲
2 p 0
␦⑀rE0* · Epdr
= 共np2 − n22兲
Fig. 3. (Color online) Plot of 兩Xlm兩2 / Wlm, which represents the
sensitivity of the resonance shift of a WGM mode with l = 500, v
= 1 to deposition of a spherical particle, as a function of ␪, the polar angle of the spot where the particle lands. The numbers adjacent to the curves are denoted 兩m兩. The mode of 兩m兩 = 500 has
the most sensitive zone at the equator of the resonator.
2
where E0 is taken at 共a , ␪ , ␸兲.
For the TE mode, use of Eq. (9) allows Eq. (34) to be
rewritten as
1385
␩ Vp 共␹l⬘/␹l兲2兩Ylm共␪兲兩2 + 共n2k0a兲−2兩Zlm兩2
Wlm a3
共␹l⬘/␹l兲2 + l共l + 1兲共n1k0a兲−2
.
共39兲
共35兲
The plot of −共␦k / k0兲TM / ␩ of a given l and m is nearly identical to that for −共␦k / k0兲TE / ␩, except for a small difference
in magnitude.
1386
J. Opt. Soc. Am. B / Vol. 23, No. 7 / July 2006
I. Teraoka and S. Arnold
共␦k/k0兲TE = − 共␩/4␲兲共Vp/a3兲N.
In the TM-to-TE shift ratio
共␦k/k0兲TM
共␦k/k0兲TE
=
共␹l⬘/␹l兲2 + 共n2k0a兲−2兩Zlm兩2/兩Xlm兩2
共␹l⬘/␹l兲2 + l共l + 1兲共n1k0a兲−2
,
共40兲
the field dependence is mostly canceled. The ratio is independent of np. Figure 4 shows the shift ratio for n1 = 1.47
and n2 = 1.33 as a function of ␪ for the same set of l, v, and
兩m兩 as those in Fig. 3. The curve for 兩m兩 = 500 is close to
horizontal for the whole range of ␪; the shift ratio is
nearly independent of the spot of adsorption. The ratio is
close to 共n1 / n2兲2 = 1.222, as 共␹l⬘ / ␹l兲2 is smaller than the
other terms in Eq. (40) and 兩Zlm兩 / 兩Xlm兩 ⬵ l for most ␪. The
curve for 兩m兩 = 499 has a dip at ␪ = 90°, but it is where the
WGM of l − 兩m兩 = 1 is insensitive to the adsorption. The
dashed portion of the curve represents the insensitive
zone where 兩X500,499兩2 / W500,499 is less than unity in Fig. 3.
The curves for 兩m兩 = 498, 497, 496 in Fig. 4 also have dips,
but they occur in the zone where 兩X500,m兩2 / W500,m ⬍ 1, indicated by dashed curves. The solid portions of the curves
for different values of 兩m兩 nearly overlap. We can conclude
that the TM-to-TE ratio’s being close to 1.2 indicates adsorption of a spherical particle, regardless of the particle’s
refractive index or the spot of adsorption. Below, we show
that the ratio is insensitive to v and l as well.
If the adsorption occurs at the equator 共␪ = ␲ / 2兲 and the
mode has 兩m兩 = l, then 兩Zlm兩 / 兩Xlm兩 = l + 1. The shift ratio
given by Eq. (40) will be nearly equal to Rl共k0a ; n1 , n2 , n2兲.
The latter Rl results because the volume fraction of the
small particle in a spherical shell at r = a+ is almost zero,
and therefore np1 = n2. Together with the absence of sensitivity of the shift ratio to ␪, we find that this Rl is not limited to adsorption onto the equator.
Now we consider random deposition of many small
spherical particles of volume Vp onto the resonator’s surface at low surface coverage. Such deposition is possible
because typically a ⬎ 100 ␮m, and many particles of
10 nm diameter, for instance, can be deposited onto the
resonator’s surface without interfering with one another.
Then the whole shift will be simply the sum of shifts by
single particles. As the average of 兩Xlm兩2 over all solid
angles is Wlm / 4␲, random deposition of N noninterfering
particles causes the TE mode to shift by
Fig. 4. (Color online) TM-to-TE shift ratio of the WGM for deposition of a single spherical particle, plotted as a function of the
polar angle for the spot of deposition. The WGM modes are l
= 500, 兩m兩 = 496– 500. The dashed curves are insensitive to the
deposition.
共41兲
Equation (41) is identical to the equation that we obtained
earlier
by
considering
simple
energy
perturbation.3 The excess polarizability of each particle is
equal to ⑀0Vp共np2 − n22兲3n22 / 共2n22 + np2兲 with a local field
correction. Note that the polarization in the particle
changes the energy in media 1 and 2 as well, but those
changes do not contribute to ␦k / k0. Likewise, the TM
mode will shift by
共␦k/k0兲TE = − 共␩/4␲兲共Vp/a2兲NRl共k0a;n1,n2,n2兲.
共42兲
The TM-to-TE shift ratio is exactly equal to
Rl共k0a ; n1 , n2 , n2兲. The N and Vp dependence disappears
in the ratio.
It is interesting to compare these shifts and their ratio
with those caused by the closest packing of identical
spherical particles on the resonator surface. We regard
this monolayer as having a uniform thickness 2b and a
refractive-index profile ␦共n2兲 that changes with r as
␦共n2兲 =
2␲
3
1/2
共np2 − n22兲
共r − a兲共2b − r + a兲
共43兲
共2b兲2
for a ⬍ r ⬍ a + 2b. We obtained this mean-field (MF) profile
by calculating the area fraction of circular cross sections
of the particles in a thin spherical shell of radius r. Applying Eq. (43) to Eq. (23) gives the MF TE shift:
冉 冊
␦k
k0
=−
TE,MF
2 ␲ n p2 − n 22 b
33/2 n12 − n22 a
共44兲
.
Likewise, the MF TM shift is calculated as
冉 冊
␦k
k0
=−
TM,MF
2n22
2
b
2
n1 − n2 a
冏
1−
n 22
BC
ln
n2
C−B
⫻Rl共k0,a;n1,n2,n2兲,
冏
共45兲
where B ⬅ 关2−13−1/2␲共np2 − n22兲兴1/2 and C ⬅ 共n22 + B2兲1/2.
If we extend Eqs. (41) and (42) to the packed layer of
particles with Vp = 共4␲ / 3兲b3 and N = 2␲ ⫻ 3−1/2共a / b兲2, the
TE and TM shifts by the noninterfering particles (NI) will
be
共␦k/k0兲TE,NI = − 共2␲/33/2兲␩共b/a兲,
共46兲
共␦k/k0兲TM,NI = − 共2␲/33/2兲␩共b/a兲Rl共k0a;n1,n2,n2兲.
共47兲
The ratio of 共␦k / k0兲TE,MF to 共␦k / k0兲TE,NI is equal to
unity at np = n2 and increases nearly linearly with an increasing np. In contrast, 共␦k / k0兲TM,MF / 共␦k / k0兲TM,NI decreases nearly linearly with np. The deviation of the MFto-NI shift ratios from unity is ascribed to the failure of
the assumption of no interference. Below we consider how
the presence of other particles nearby changes the resonance shift.
In the TM polarization, E02 is approximately perpendicular to the surface on which particles sit. The dipolar
field by a particle will weaken the field at nearby particles, which decreases the perturbation integral. This effect is stronger for particles of a greater np. Therefore the
MF-to-NI shift ratio is less than unity and decreases with
increasing np.
I. Teraoka and S. Arnold
Vol. 23, No. 7 / July 2006 / J. Opt. Soc. Am. B
1387
to uniform electric field E02, electric field Ep within the
cylinder is uniform and is given as16
Ep =
Fig. 5. (Color online) TM-to-TE shift ratio for monolayer coverage by spherical particles plotted as a function of refractive index
np of the particles for n1 = 1.47, n2 = 1.33 and l = 500. Solid and
dashed curves are for v = 1 , 2. For reference, the ratio for l = 150,
v = 1 is shown as a dashed–dotted curve. The horizontal lines are
the ratios when a single spherical particle is adsorbed.
2np2
n 22 + n p2
E02 .
共48兲
It can be shown that the integral of ␦⑀rE0* · Ep over the
particle volume is not affected by the presence of the resonator for TE and TM modes in the same way as was demonstrated in Appendix B for a spherical particle. Therefore we can use Eq. (48) for the integral. When the
cylinder axis is parallel to E02, Ep = E02.
Now the particle is anisotropic. The shift ratio depends
on the orientation of the cylinder. Below, we show the results for the TM-to-TE shift ratio for many cylinders lying
on the resonator’s surface and cylinders standing vertically on the surface. We assume random distribution at a
low coverage. For the so-called lying cylinders, a random
orientation on the surface is also assumed.
For cylinders lying on the surface, the TM-to-TE shift
ratio is given as
共␦k/k0兲TM
共␦k/k0兲TE
=
共␹l/␹l兲2 + 关l共l + 1兲/共n2k0a兲2兴4n22/共3n22 + np2兲
共␹l/␹l兲2 + 关l共l + 1兲/共n1ka兲2兴
.
共49兲
For cylinders standing on the surface, the ratio is
共␦k/k0兲TM
共␦k/k0兲TE
=
共␹l/␹l兲2 + 关l共l + 1兲/共n2k0a兲2兴共n22 + np2兲/2n22
共␹l⬘/␹l兲2 + 关l共l + 1兲/共n1ka兲2兴
.
共50兲
Fig. 6. TM-to-TE shift ratios for deposition of a spherical particle (short-dashed curve) and a cylindrical particle (lying and
standing cylinders) and adsorption of a packed layer of spherical
particles plotted as a function of np, the refractive index of the
particles. Parameters used are n1 = 1.47, n2 = 1.33, l = 500, v = 1.
In the TE polarization, E02 is parallel to the surface.
Depending on the relative position, the dipolar field by a
particle will strengthen or weaken the field at nearby particles. When np ⬎ n1, the net result is strengthening. The
effect becomes stronger with increasing np.
Figure 5 shows how the TM-to-TE ratios shift for deposition of a single particle and the deposition of a packed
layer of particles changes with np for l = 500, n1 = 1.47, and
n2 = 1.33. For the single particle, m = 500, and deposition
onto the equator is assumed. The ratio is shown for two
mode orders, v = 1 , 2. For reference, the ratio is also shown
for l = 150 and v = 1. Both ratios depend only weakly on v
and l. The ratio for the monolayer coverage is equal to
Rl共k0a ; n1 , n2 , n2兲 when np = n2 and decreases with increasing np. The latter decrease indicates that the TMto-TE ratio will drop from ⬃1.2 to a value that depends on
np as more particles are adsorbed onto the resonator. The
large drop in the shift ratio is caused by the dipolar fields
of nearby particles, as discussed above.
It is interesting to compare the TM-to-TE shift ratio for
adsorption of a single cylindrical particle with the ratios
that we obtained so far. When a cylinder of uniform refractive index np is placed in the direction perpendicular
Figure 6 compares the shift ratios for the two orientations of the cylinder with the shift ratio for a spherical
particle and that for a packed monolayer of the spherical
particles. The parameters used are n1 = 1.47, n2 = 1.33, l
= 500, and v = 1. When np = n2 (outside the plot range), the
ratio is equal to Rl共k0a ; n1 , n2 , n2兲 for all particle geometries. The shift ratio for the lying cylinder decreases with
an increasing np. Such a cylinder is essentially an isolated
spherical particle in the direction perpendicular to the
cylinder axis and is an extended medium in the axis’s direction. It is then reasonable that the curve for the lying
cylinder runs between the curve for the isolated spherical
particle and the curve for the packed spheres. What is interesting is the curve for the standing cylinder, which increases with increasing np. The standing cylinder has
Ep ⬵ E02 in the TM polarization, whereas Ep ⬍ E02 in the
TE polarization, thereby making the TM-to-TE shift ratio
greater than unity.
4. CONCLUSIONS
Using the boundary conditions of the electromagnetic
waves, a general formula was derived for the resonance
wave vector shift of a whispering gallery mode in a transparent spherical resonator by a perturbation in the
refractive-index profile in the surrounding medium. The
formula allows for a locally large change in the refractive
index such as a change caused by deposition of a particle.
The formula for the first-order perturbation was applied
to adsorption of a thin layer and deposition of a single
1388
J. Opt. Soc. Am. B / Vol. 23, No. 7 / July 2006
I. Teraoka and S. Arnold
spherical particle and a cylindrical particle. It was found
that the resonance shifts of the TE and TM modes and
their ratio sensitively reflect the shape and state of the
adsorbate.
The results for a packed layer of spherical particles are
based on a mean-field refractive index profile. We are preparing a manuscript for the TE and TM shifts that take
into account the dipolar fields by neighboring particles explicitly at different surface densities of the particles.17
To calculate the volume integral on the left-hand side of
Eq. (5), we divide V into V1, Vp, and the rest V2. We denote by Ei 共i = 1 , 2 , p兲 the perturbed electric field in the respective space and by E0i 共i = 1 , 2兲 the unperturbed field.
In V1 and V2, E01 and E02 change to E1 and E2, respectively, but the changes are small. The change from E02 to
Ep in Vp, however, may be large enough to satisfy the
boundary condition on the surface of Vp. Therefore we
cannot simply replace E with E0 in Eq. (5). The latter is
allowed only when ␦⑀r is small everywhere.10
We consider a three-part integral I, defined as
兺
i=1,2,p
冕
Vi
The volume integral can be converted to a surface integral
by application of the following identity in each of V1, Vp,
and V2:
共E0* · ⵜ ⫻ ⵜ ⫻ E − E · ⵜ ⫻ ⵜ ⫻ E0*兲dr
Vi
=
冕
dS · E02* ⫻ ⵜ ⫻ 共Ep − E2兲.
共A4d兲
Sp
In Eq. (A4a), the Maxwell equations applied E01* and
E02* in the respective spaces give ⵜ ⫻ E0i* = −i␻0B0i* 共i
= 1 , 2兲, where B0i* is the complex conjugate of B0i, the unperturbed magnetic flux density in volume i. By simply
writing B0 for B0i, we have
冕
dS · 共E1 − E2兲 ⫻ B0* .
共A5兲
S1
From continuity of the tangential component of E across
the interface, we find that only the normal component exists in E1 − E2. Since 共E1 − E2兲 ⫻ B0* is perpendicular to
dS, I1 = 0. Likewise, we can show that I2 = 0 and I3 = 0. In
Eq. (A4d), ⵜ ⫻ 共Ep − E2兲 = i␻共Bp − B2兲, where ␻ = ck. Continuity leads to Bp = B2 on Sp. Therefore I4 = 0. Thus we
have I = 0.
APPENDIX B
共E0* · ⵜ ⫻ ⵜ ⫻ E − E · ⵜ ⫻ ⵜ ⫻ E0*兲dr.
共A1兲
冕
冕
I1 = − i␻0
APPENDIX A
I⬅
I4 =
dS · 共E ⫻ ⵜ ⫻ E0* − E0* ⫻ ⵜ ⫻ E兲 共i = 1,2,p兲,
Si
The purpose of this appendix is to prove Eq. (34). Let z be
the direction of E02, the field exterior to the resonator, before the spherical particle of radius b is placed; see Fig. 7.
Since b Ⰶ ␭ Ⰶ a, E02 can be regarded as static and spatially
uniform (quasi-static approximation11), and the resonator
surface as planar. We do not specify where the resonator
(medium 1) is. We show that its location is irrelevant to
the perturbation integral.
We set up the spherical polar coordinate 共r , ␪ , ␸兲 with
the origin at the center of the particle and the polar axis
in the z direction. This coordinate is different from the
one used for the resonator in the main part of this paper.
The potential ⌽P within the spherical particle is expressed as
共A2兲
where Si denotes the surface of Vi and the surface normal
dS points from the interior of Vi to the exterior. The surface of S2 consists of two interfaces: At the interface to V1,
dS for S2 is equal to −dS of S1. At the interface to Vp, dS
for S2 is equal to −dS of Sp. Therefore Eq. (A2) is rewritten as
共A3兲
I = I1 − I2 + I3 − I4 ,
⌽P共r, ␪, ␸兲 = ⌽P0 +
兺A
l
r Ylm共␪, ␸兲,
m l
共B1兲
l,m
where ⌽P0 and Alm are constants, l = 1 , 2 , . . ., m = −l, −l
+ 1 , . . . , l, and Ylm共␪ , ␸兲 is a spherical harmonic function.
The z component of the field within the spherical particle,
EPz, is expressed as
where the surface integrals on S1 and Sp are given as
I1 =
冕
dS · 共E1 ⫻ ⵜ ⫻ E01* − E2 ⫻ ⵜ ⫻ E02*兲, 共A4a兲
冕
dS · 共E01* ⫻ ⵜ ⫻ E1 − E02* ⫻ ⵜ ⫻ E2兲, 共A4b兲
S1
I2 =
S1
I3 =
冕
Sp
*
dS · 共Ep − E2兲 ⫻ ⵜ ⫻ E02 ,
共A4c兲
Fig. 7. (Color online) Quasi-static TE field Ep in the microsphere. The fields far from the sphere are E01 and E02, respectively, in media 1 and 2. The gray (orange online) hatching represents the top surface of 1 in the TM mode.
I. Teraoka and S. Arnold
EPz = −
Vol. 23, No. 7 / July 2006 / J. Opt. Soc. Am. B
兺A
l
兺A
l
m
REFERENCES
lrl−1 cos ␪Ylm共␪, ␸兲
1.
l,m
m l−1
sin ␪
⳵
Ylm共␪, ␸兲.
共B2兲
2.
The orthogonality of the spherical harmonics gives the
perturbation integral as
3.
+
r
l,m
冕
⳵␪
E02* · Epdr = − 共4␲/3兲1/2b3E02A10 .
共B3兲
4.
Vp
Note that only the rY10 component in ⌽P, which generates
a uniform field in the sphere, is relevant to the integral.
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we write ⌽2 as
⌽2共r, ␪, ␸兲 = − 共4␲/3兲1/2E02rY10共␪, ␸兲 +
兺
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5.
6.
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l,m
共B4兲
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冕
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Vp
3n22
2n22 + nP2
VpE022 ,
1389
共B5兲
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show that medium 1 is irrelevant to the integral.
8.
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11.
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16.
ACKNOWLEDGMENT
This research was supported by NSF through Division of
Bioengineering and Environmental Systems grant
0522668.
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