February 15, 2003 / Vol. 28, No. 4 / OPTICS LET TERS 272
Shift of whispering-gallery modes in microspheres by
S. Arnold, M. Khoshsima, and I. Teraoka
Microparticle Photophysics Lab (MP3L), Polytechnic University, Brooklyn, New York 11201
Los Gatos Research, Mountain View, California 94041
Center for Studies in Physics and Biology, Rockefeller University, New York, New York 10021
Received July 16, 2002
Biosensors based on the shift of whispering-gallery modes in microspheres accompanying protein adsorption
are described by use of a perturbation theory.For random spatial adsorption, theory predicts that the shift
should be inversely proportional to micorsphere radius R and proportional to protein surface density and
excess polarizability. Measurements are found to be consistent with the theory, and the correspondence
enables the average surface area occupied by a single protein to be estimated.
with crystallographic data for bovine serum albumin.
is found to be extremely sensitive to the target region, with adsorption in the most sensitive region varying
as 1?R5?2.Specific parameters for single protein or virus particle detection are predicted.
Society of America
T hese results are consistent
T he theoretical shift for adsorption of a single protein
© 2003 Optical
In therecent past, theneed for miniaturebiosensors for
the detection of infectious agents, toxins, protein, and
DNA has taken on added urgency as the world antici-
pates further bioterrorism.
reported specific detection of unlabeled biomolecules
on a spherical surface (radius R ?0.15 mm) from the
frequency shift of whispering-gallery modes (WGMs).
The modes were stimulated in a dielectric sphere
immersed in an aqueous environment by means of cou-
pling light evanescently from an optical fiber.2
authors claimed unprecendented sensitivity for the
adsorption of protein molecules with spatial unifor-
mity.In what follows, we (1) introduce an optical
theory that describes this effect in an asymptotic
limit ?2pR?l . .1?, (2) compare the predicted size
dependence with that from new experiments, and
(3) calculate the effect of reducing the size while
placing protein molecules at specific locations on
the sphere surface. We show that, for particular
locations, the sensitivity to single-protein adsorption
can be enhanced by orders of magnitude.
Figure 1 illustrates the basic configuration of inter-
est.Light from a tunable distributed feedback laser
is coupled into a WGM of the sphere from an eroded
optical fiber3and circulates about the equator.
nant modes are detected from dips in the transmis-
sion through the fiber. A protein molecule diffuses
to the sphere’s surface from the surrounding aqueous
medium and is adsorbed at position ri, where it in-
teracts with the evanescent field of the WGM.
index i distinguishes each adsorbed protein molecule.
This interaction polarizes the molecule, shifting the
frequency of the mode.
To evaluate the shift dv in angular frequency v of
a single protein molecule, it is useful to consider the
Recently Vollmer et al.1
energy of interaction as a first-order perturbation to
a single-photon resonant state, with semiclassical field
E0?r?eivt. The evanescent tail of the field induces a
dipole moment in the protein in excess of the displaced
water, dpeivt, causing a shift in the photon energy
of the resonant state, ¯ hdv ? 2dp ? E0??ri??2. The
excess dipole moment can be represented in terms of
the real part of an excess polarizability aex, i.e., dp ?
aexE0?ri?.The fractional frequency shift for a protein
positioned at riis given by the result of dividing the
perturbation by the energy of the mode (i.e., ¯ hv), as
represented by integrating over the energy density in
surface of a sphere near an eroded optical fiber core.
sphere and fiber are surrounded by an aqueous solution.
Nanoscopic protein molecule at position ri on the
0146-9592/03/040-03$15.00/0© 2003 Optical Society of America (ms #16351a(tmp))
273 OPTICS LET TERS / Vol. 28, No. 4 / February 15, 2003
The integral in the denominator is taken over the
interior of the sphere, which contains the overwhelm-
ing majority of the mode energy ?.94%?.4
approximation simplifies the analysis by allowing
the homogeneous permittivity, es, of the sphere to be
pulled through the integral.
this integral results from adding equal electric and
It should be noted that for protein molecules, which
arecomposed of a variety of aminoacids, aexis roughly
proportional tothe mass of the molecule,5and the shift
in frequency in accordance with relation (1) should be-
have in the same way.
Relation (1) represents the shift that is due to an
individual molecule at an arbitrary position on the
sphere, a point we will return to in calculating the op-
timal effect.However, in Ref. 1 it was reported that a
large number of protein molecules are distributed over
random locations on the sphere’s surface.
for all these molecules we sum the singular contribu-
tion in relation (1) over N randomly located molecules
and then turn this discrete sum into an integral over
where sp, the protein surface density, is N??4pR2?.
With this transformation from a discrete tocontinuous
sum, relation (1) becomes
The factor of 2 preceding
where esis written in terms of a relative permittivity,
We now evaluate relation (2) for a general TE mode
for which the interior field at distance r from the
sphere center is given as E0? Ainjl?k0rpers?ˆLYlm,6
where Ainis the amplitude, jl?z? is a spherical Bessel
function, ˆL is a dimensionless angular momentum
operator ?ˆL ? 2ir 3 =?, k0? v?c with c being the
speed of light in vacuum, and Ylm is a spherical
harmonic function.Fortunately both the surface and
volume integrals in relation (2) contain precisely the
same angular integrands.Consequently,
where R is the radius of the sphere.
nance, the volume integral in the denominator of
relation (3) may be asymptotically ?2pR?l ..1?
ers?, where erm is the relative permittivity of the
Inserting this expression into
relation (3), we find that the fractional frequency shift
is given by a surprisingly simple formula:
0?jl?k0rpers??2r2dr ?R3/2?jl?k0Rpers??2??ers2 erm??
e0?ers2 erm?R? 2
wherensand nmaretherefractiveindices of thesphere
and the aqueous medium, respectively.
of dv?v for TM modes involves changing the field in
relation (2).Theresult produced by a similar analysis
has the same aexsp?R dependence with numerically
calculated shifts that only differ from the TE shifts by
a few percent for our silica–water interface.
The 1?R size dependence in relation (4) is expected
for a homogeneous sphere.
a layer that is dR thick, it must preserve the prod-
uct k0R for a given resonance, and consequently
dk0?k0 ? dv?v ? 2dR?R. However, the formula
becomes more complicated when the sphere is optically
heterogeneous, as revealed in relation (4).
less, when the surface is saturated with protein,
as revealed by no additional shift regardless of the
external concentration, a plot of 2dv?v versus 1?R
will have slope dReff, the effective thickness of the
layer.It should be noted that dReffas defined can be
negative, if the absorbed material has a polarizability
less than that of an equal volume of water.
circumstance is not the case for protein adsorption,
since the optical permittivity of proteins is higher than
that of water.In fact, proteins have permittivities
close to that of quartz.
We have performed experiments on the adsorption
of bovine serum albumin (BSA) protein on quartz
microspheres. The silica glass surface is sensitized
for protein adsorption by chemical modification with
vapor-phase 3-aminopropyltriethoxysilane following
2dv?v measured for complete saturation by use of
a current-tuned distributed feedback laser9operating
at a nominal wavelength of 1.34 mm, are shown as
a function of 1?R in Fig. 2.
implemented only after equilibrium was reached at
23±C. The system was verified to have returned to
this temperature when wavelength-shift measure-
ment was taken.The spheres ranged in radius from
88 to 232 mm ?412 , 2pR?l , 1087?.
scatter in the data over this size range, a 1?R size
dependence appears reasonable.
is dReff? 3.6 nm.
An effective thickness of 3.6 nm is very close to the
smallest dimension of BSA as revealed through x-ray
If such a sphere accretes
Protein injection was
The slope of the fit
BSA protein adsorption versus 1?R.
based on relation (4), which gives a surface density sp?
2.9 3 1012cm22.
Saturation shifts of WGM resonances measured for
T he solid line is a fit
February 15, 2003 / Vol. 28, No. 4 / OPTICS LET TERS 274 Download full-text
with a heart-shaped profile.
sion is the height of the pancake.
from the effective thickness and relation (4), it is
possible to estimate the molecular surface density,
sp? dReffe0?ns22 nm2??aex.
face density by use of the excess polarizability arrived
at from differential refractive-index measurements1
?aex ? 4pe0?3.85 3 10221cm3??, and the usual re-
fractive indices for quartz and water, with the result
sp ? 2.9 3 1012cm22. So a BSA molecule occupies
an area sp21? 3.4 3 10213cm2.
again with crystallographic data, for which the area
of the heart-shaped projection is 3.7 3 10213cm2.
appears that BSA forms an extremely compact layer
on the microsphere surface.
Finally, we are interested in the possibility for
would be possible by looking at steps in the change
of dv?v with time, and this in turn provides a
possible means for separately measuring aex.
the light within a WGM circumnavigates the equator
?u ? p?2? in an orbit that is confined to a thin ring,
molecules at polar angles outside the ring cannot
influence the mode frequency.
comes from molecules that stick at
TE mode that circulates at the equator l ? m, and
the angular intensity is proportional to jˆLYllj2, which
for large l is proportional to jYllj2.11
the frequency shift for a protein at the equator to
that averaged over random positions on the surface
is enhanced by a factor EF ? 4pjYll?p?2, w?j2.
spatial enhancement EF
the average size particle used in Fig. 2, l ? 1000
and EF ?36. To obtain the average shift for an
individual protein at a random position, we set the
surface density in relation (4) to s ? 1??4pR2? with
the result ?dv?v?r? 2aex??4pe0?ns22 nm2?R3?.
shift that is due to a single protein at the equator is
?dv?v?e? EF 3 ?dv?v?r, or
BSA resembles a thick pancake
The smallest dimen-
We calculate this sur-
This agrees well
The greatest signal
u ? p?2.For a
So the ratio of
Forcan be significant.
This single-protein shift has a large size dependence.
Since jYll?p?2, w?j2increases roughly in proportion to
l1?2or R1?2, the single-protein shift should goas R25?2.
Currently, we can detect a fractional frequency change
as small as 1028. Since we can see a shift of one
fiftieth of a linewidth, this requires that Q be 2 3 106.
This value of Q is controlled by overtone vibrational
absorption of water at 1.34 mm and the size of the
microsphere.Leakage at the quartz–water interface
limits the smallest radius for which this sensitiv-
ity is reasonable to approximately 50 mm.
first-order TE mode within such a particle, and for
a wavelength of 1.34 mm, 4pjYll?p?2, w?j2? 20.8.
Under these conditions, the smallest detectable single-
protein polarizability is asd? 4pe0?2.4 3 10217cm3?,
or 6230 times the polarizability of BSA.
masses seldom exceed 106Da, which is only 15 times
the mass of BSA. Thus single-protein measurements
are unrealistic from the resonance shift at 1.34 mm.
We may overcome the problem by working in the
blue, where water absorption is reduce by more than
a factor of 100, and by choosing a material for the
microsphere with a larger refractive index.
As an example, a microsphere of amorphous sap-
phire has a refractive index of 1.7 at a wavelength of
400 nm (blue diode laser with external cavity), which
enables the radius to be reduced to approximately
3.6 mm for Q of 2 3 107in water.
we are able to see a shift of a fiftieth of a linewidth
as before, the least measurable fractional shift would
be 1029. The minimum detectable polarizability pro-
jected from Eq. (5) is now approximately three times
the polarizability of BSA, a number that is consistent
with large protein molecules such as thyroglobulin,
ferritin, and virus particles (e.g., lambda phage).
Adsorption onto the equator may be promoted by
selective silanization of the equator.
Research at the Polytechnic was supported by a
National Science Foundation grant (BES-0119273).
F. Vollmer was supported by a fellowship of the
address is firstname.lastname@example.org.
*Present address, Sandia National Laboratories, Al-
buquerque, New Mexico 87185.
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