Capillary-Type Microfluidic Sensors Based on Optical ...

Delivered by Publishing Technology to Editor in ChiefIP 12910778221 On Tue 09 Dec 2014 222037

Copyright American Scientific Publishers

Copyright copy 2014 by American Scientific Publishers

All rights reserved

Printed in the United States of America

Reviews in Nanoscience and NanotechnologyVol 3 pp 193ndash209 2014(wwwaspbscomrnn)

Capillary-Type Microfluidic Sensors Based onOptical Whispering Gallery Mode ResonancesA Meldrumlowast and F Marsiglio

Department of Physics University of Alberta Edmonton AB T6G2E1 Canada

Microfluidic capillary-type sensors could potentially find applications in a range of medical and environmentaldiagnostic technologies Refractive-index sensors-as well as biosensors-based on the whispering gallery moderesonances of a thin-walled capillary offer several modes of operation and the capacity to detect a wide rangeof analytes through standard silica surface functionalization methods This article reviews the basic opticalphysics associated with the resonances and how they respond to changes in the local refractive index Thesensitivity and detection limits of these devices are explored and at least one old assumption concerning theinverse relationship between the quality factor of the resonance and the detection limit is shown to be validonly under a restricted set of circumstances Problems and pitfalls associated with whispering-gallery-modecapillary sensors are examined as well as the experimental methods needed for analysis Finally relevantcapillary sensor examples from the literature are summarized

KEYWORDS Liquid Core Optical Ring Resonator Fluorescent Core Microcapillary Capillary Sensor MicrofluidicWhispering Gallery Modes

CONTENTS1 Introduction 1932 Historical Background 1943 Cylindrical WGMs Basic Concepts 1954 Mode Characteristics 1995 Sensitivity and Detection Limits 200

51 Calculating the Refractometric Sensitivity 20052 Detection Limits Getting the Most

Out of the Measurement 2016 Experimental Techniques (Fabrication Functionalization

Excitation and Analysis) 20161 Sample Fabrication 20162 Setup 20263 Data Collection 202

7 Pitfalls 20271 Thermal Effects 20272 Mechanical Drift 20373 Spiral Modes 204

8 Examples of Relevant Sensors 2049 Conclusions and Outlook 206

Acknowledgments 206References and Notes 206

1 INTRODUCTIONCapillaries have been widely investigated for their manychemical and biochemical sensor-related applications1ndash3

lowastAuthor to whom correspondence should be addressedEmail ameldrumualbertacaReceived 21 December 2013Accepted 28 June 2014

A variety of sensing transduction techniques can beemployed and have been reviewed45 including capillaryelectrophoresis6ndash10 fiber-optic methods that couple lightinto and out of a capillary11 (eg Bragg gratings) andsimple optical refractive index sensors1213 Microfluidicoperation can be especially interesting for the developmentof small parallelized devices or for situations in which thesample volume is small14ndash16

Thin-walled glass capillaries represent one promisingavenue toward the development of microfluidic optical sen-sors The outer wall traps light by total internal reflectionas it circulates around the capillary circumference whilefluids can be pumped into the channel region The cylindri-cal electromagnetic resonances that form the basis of thetransduction mechanism are called the whispering gallerymodes (WGMs)WGMs have been the subject of several recent reviews

that have focused mainly on the properties of opti-cal microspheres17ndash19 Optical resonances in microdisks20

and pillars21 have also been widely investigated andreviewed partly due to the ability to control the spon-taneous emission rate for atoms or quantum dots placedin the cavity22ndash26 The cylindrical WGMs have also beenreviewed2728 although mostly without a sensor29 focusThe intent of this work is to overview the work so faron capillary WGM-type sensor structures and to outlinethe basic physics and experimental issues that would behelpful for researchers wishing to enter the field

Rev Nanosci Nanotechnol 2014 Vol 3 No 3 2157-936920143193017 doi101166rnn20141054 193



Capillary-Type Microfluidic Sensors Based on Optical Whispering Gallery Mode Resonances Meldrum and Marsiglio

2 HISTORICAL BACKGROUNDThe term ldquowhispering galleryrdquo seems to have originated todescribe the peculiar acoustic effects that were reported indomed buildings such as St Paulrsquos cathedral in London30

and the Gol Gumbaz Mausoleum in northern India31

In these circular domed structures a whisper at one sideis clearly audible at the other side of the dome but itcannot be heard near the center Lord Rayleighrsquos 1910paper titled ldquoThe problem of the whispering galleryrdquo30

specifically focused on what he called an acoustic ldquocling-ing effectrdquo Essentially the ldquosonorous vibrationsrdquo producednear one wall of the cathedral appeared spatially limitedto a tiny fraction of the radius very near the cathedralwall at the other side (as measured by the flicker of acandle flame) Lord Rayleighrsquos paper reported that theBessel function Jlz which was the known solution fora circular resonance with fixed boundaries has interestingproperties if l is large In such cases he showed that theresonant vibrations must indeed ldquoclingrdquo very near to theouter radius of the resonator and are nearly zero insideexactly as observedGustav Mie had just a few years earlier developed

the theory for the scattering of light by small aerosoldroplets32 These structures were known to have electro-magnetic resonances that were similar in many respects toLord Rayleighrsquos whispering gallery modes and the nameeventually referred to both the acoustic and optical reso-nances Part of Miersquos work was based on earlier light scat-tering studies by Thomson33 Alfred Clebsch34 LudwigLorenz3536 and Peter Debye37 A complete history ofMiersquos electromagnetic scattering theory was assembled by

A Meldrum obtained his PhD in 1997 from the University of New Mexico where hestudied radiation-solid interactions in materials for nuclear waste disposal He went on toa postdoctoral fellowship at Oak Ridge National Laboratory working on the microstruc-ture and properties of semiconductor and magnetic nanocrystals Dr Meldrum joined theUniversity of Alberta as Assistant Professor in physics in September 1999 Currently hefocuses on optical cavities optical sensors fluorescence imaging and spectroscopy and thebasic physics of cavity-quantum-dot interactions He additionally works on new ways tobring research into the classrooms and to provide undergrads a greater exposure to physicsresearch Now a full professor and Vargo Teaching chair he has more than 130 refereedpublications in physics materials science optics and earth science journals

F Marsiglio graduated with a BASc from the Engineering Science program at the Uni-versity of Toronto in 1983 then went on to do graduate work in physics at McMasterUniversity where he received both a Masters and PhD (1988) Subsequently he held anNSERC postdoctoral fellowship at the University of California San Diego In 1990 hejoined AECL Research at Chalk River Laboratories During this time his work has focusedon the theory of superconductivity particularly as applied to the so-called high temperaturesuperconductors In 1997 he became an associate professor in the Department of Physicsat the University of Alberta and a full professor in 2001 Coincidentally he became theDirector of the Theoretical Physics Institute and kept this position until 2008 In 2013 hebecame Associate Chair of Research in the department He was a member of the QuantumMaterials Program in the Canadian Institute for Advanced Research His research coverssuperconductivity polarons non-equilibrium magnetism and resonances in optical cavities

Horvath38 and by Wriedt39 and these are excellent sourcesfor a more complete history of the WGMs Essentiallywhile the basic properties of optical WGMs were under-stood in the early part of the 20th century the mathematicsdescribing the Bessel functions needed in the Mie the-ory were for the most part too complicated to be fullytreated until the advent of computers Much more recentefforts gave complete modern descriptions of the sphericalWGMs4041

Cylindrical structures have a set of resonances that aresimilar to the spherical ones the main difference beingthat there is no polar component to the field which can beassumed to propagate in a 2-dimensional cylindrical crosssection similar to Rayleighrsquos acoustic modes (the effect ofwaves with some component of their wavevector along thecapillary axis will also be discussed however) The equa-tions used to model the resonances are mainly the same asfor the spherical case with the main differences being theuse of cylindrical rather than spherical Bessel functionsand the lack of a ldquopolarrdquo direction (ie one need not con-sider the Legendre harmonics) in the field solutionsThus only a few parameters are needed to completely

describe the capillary resonances the number of inten-sity maxima in the radial direction (n) the number ofwavelengths that ldquofitrdquo around the cylinder for a given res-onance (l) and the quality (Q) factor that describes howwell the mode is able to trap radiation Three such reso-nances are illustrated in Figure 1 showing different val-ues of n The polarization can be either transverse electric(TE) or transverse magnetic (TM) Unfortunately thereis no consensus as to the definition of the polarization

194 Rev Nanosci Nanotechnol 3 193ndash209 2014



Meldrum and Marsiglio Capillary-Type Microfluidic Sensors Based on Optical Whispering Gallery Mode Resonances

Fig 1 Electric field amplitude for the first three transverse electricradial mode orders n = 123 of a disk All three resonances have thesame angular mode order l= 20 and are TE polarized The disk boundaryis denoted by the thin green line

Fig 2 Sketch of the LCORR (left) and FCM (right) structures In bothdevices m1 is the analyte refractive index and m3 is the outside index (airin the LCORR case the glass capillary wall in the FCM) The confininglayer has index m2 it is comprised of the thinned capillary wall (in theLCORR) or a thin fluorescent layer (for the FCM) which can typically bemade of quantum dots (the dotted layer in the right-hand diagram) Theelectric field profile of a resonance is sketched superimposed onto thediagram (the black ldquowaverdquo) In the case of the FCM the outer capillaryboundary is mostly irrelevant since it is typically located very far fromthe fluorescent film and channel regions

directions for capillaries We prefer to define the polariza-tion with respect to the WGM propagation plane thus TEresonances have the electric field parallel to the capillaryaxis whereas TM modes have the magnetic field parallelto the axis

The two main types of device structures are the liquidndashcore optical ring resonator (LCORR) and the fluorescentcore microcapillary (FCM) Both are three-layer cylindri-cal cavity structures (Fig 2) In the case of the liquidndashcore optical ring resonator the light-confining region(index m2) is the thinned wall of a glass capillary and theouter region (m3) is air In the case of the FCM the con-fining region is a high-index fluorescent film and the outerregion is the thick wall of the capillary In order to confineradiation into the cylindrical resonances of either structurem2 must be larger than m3 (unless layer 2 is very thinin which case m1 should also be greater than m3) Theanalyte (m1) is in the channel region for both structuresAs illustrated in Figure 2 the FCM is a larger and morephysically robust device

3 CYLINDRICAL WGMs BASIC CONCEPTSHere we are interested in solving the electric field dis-tribution for the WGM resonances of a layered cylinder

The purpose of this section is not a rigorous re-derivationof the cylindrical resonances which can be found in differ-ent formats from several sources (eg Refs [42ndash44]) butis instead aimed toward a practical and more easily under-standable method to solve the electric field resonanceswhich that can be readily implemented The solutions per-mit a straightforward calculation of the mode wavelengthQ factor and the refractometric and thermal sensitivity forLCORRs (which are synonymous with opto-fluidic ringresonators or ldquoOFRRsrdquo) fluorescent core microcapillar-ies (FCMs) or other layered cylindrical structures such ascoated optical fibers45 or microdisks We will examine a3-layer system since the two main types of experimentalrealization are effectively comprised of only three layersa channel analyte a confining region responsible for thedevelopment of the WGMs and an outside region How-ever the theory can be extended to four or more layersrelatively easilyWe review the TE polarization first since it is sim-

pler The E-field is by definition parallel to the capillaryaxis The radial and azimuthal components of the field arezero and the axial part is usually solved by separation ofvariables

Ezr= Ezr expplusmnil (1)

Here r and are the cylinderrsquos radial direction andazimuthal angle respectively and the axial direction willbe z The electric field has been separated into a radialpart Ezr and an azimuthal part eil the latter impliesthat the field will return to the starting point in phase aftermaking l oscillations (note that use of integerm rather thanl would be more consistent with respect to a derivationof the spherical resonances but here m is reserved forthe refractive index and n for the radial mode number)Other periodic functions could be used in Eq (1) but eil

represents a simple solution for an azimuthally uniformcylindrical structure The time dependence of the field willbe described laterAccording to Maxwellrsquos equations the TE resonances of

a dielectric cylinder must satisfy

2Ezr+

z

(Ezr

1

r

z

)+k20m

2Ezr= 0

(2)where is the electric permittivity of the material in whichthe field is propagating k0 is the vacuum wavevector andm is the refractive index and Ez is the electric field polar-ized along the capillary axis The second term is zerobecause the index of refraction is taken to be constant inthe z-direction The Laplacian operator in cylindrical coor-dinates is

2Ezr =2Ezr

z2+ 1

r22Ezr

2+ 1

r

r

times(rEzr

r

)(3)

Rev Nanosci Nanotechnol 3 193ndash209 2014 195




The first term on the right-hand side is also zero sincewe assume no axial (z) variation of the electric field in thecylinder which simplifies the analysis The second term is

1r2

2Ezr

2= 1

r22Ezre

il

2=minus l2

r2Ezr (4)

and the third term can be written

1r

r

r middot Ezr

r= 2Ezr

r2+ 1

r

Ezr

r(5)

Combining Eqs (2)ndash(5) multiplying both sides by r2and separating variables according to Eq (1) gives

eplusmnil

[r2d2Ezr

dr2+ r

dEzr

dr+ k2r2m2minus l2Ezr

]= 0

(6)The radial part of Eq (6) can be rewritten in the stan-

dard form using R=mkr and dR=mk middotdr

R2 d2EzR

dR2+R

dEzR

dR+ R2minus l2EzR= 0 (7)

This is the Bessel differential equation with solutionsfor EzR given by linear combinations of JlR andYlR ie cylindrical Bessel functions of the first andsecond kind of order l The cylindrical Hankel functionsH

1l R= JlR+ iYlR and H

2l R= JlRminus iYlR are

also solutions of Eq (7) and will be important for theradiating fields as described further belowFor the electric field resonance in a layered cylinder we

need to solve Eq (7) when there are piecewise disconti-nuities in the refractive index such that

mr=

⎧⎪⎪⎨⎪⎪⎩

m1 r le b

m2 b lt r le a

m3 r gt a

⎫⎪⎪⎬⎪⎪⎭

(8)

Here a is the radius of the outer boundary of the layeredcylinder and b= aminus t is the radius of the inner one wheret is the confining layer thickness For an FCM t will rep-resent the thickness of the fluorescent layer and for anLCORR it is the thickness of the capillary wallThe solution for EzR in each of the three regions

depends on the boundary conditions For the inner layer(layer 1) the requirement is that the field not be infinite atthe origin so the Bessel function of the first kind is the onlysolution that is EzR = AlJlR In the outermost layerthe waves should be traveling outwards For large argu-ment mkr an asymptotic expansion of the Hankel functionyields

H1l mkr =

radic2

mkreimkrminusl2minus4 (9)

which represents an ldquooutgoingrdquo wave in the time-dependent wave equation This implies that the field in

the outermost layer is described by EzR =DlH1l R =

DlJlR+ iYlR where Dl is another constantIn the middle layer (layer 2) we have both incoming

and outgoing waves The incoming waves are described byH

2l R= JlRminus iYlR which leads to the same expan-

sion as shown in Eq (9) except with the opposite signin the exponential term Thus the field within layer 2 canbe described as EzR= BlH

2l R+ClH

1l R where Bl

and Cl are also proportionality constants The constantsAlBlCl and Dl must be set to enforce continuity of thefield across the various boundaries given by

SR=

⎧⎪⎪⎨⎪⎪⎩

AlJlm1k0r r le b

BlH2l m2k0r+H

1l m2k0r b lt r le a

DlH1l m3k0r r gt a

⎫⎪⎪⎬⎪⎪⎭(10)

where we have set Cl = 1 and recall that R=mkr A plotof SR is the same as a radial amplitude plot of EzRThus for the TE polarization we have the following solu-tions for a dielectric cylinder

ErR = 0

ER = 0 (11)

EzR = SR

We next write some fairly simple equations for thefields which are given by the electromagnetic boundaryconditions at each dielectric interface These conditionsstate that the Ez Hr and H fields (for the TE polariza-tion) must be continuous Since the electric field Ezr hasto be continuous across the boundaries we obtain

boundary 1 AlJlm1k0b

=BlH2l m2k0b+H

1l m2k0b (12a)

boundary 2 BlH2l m2k0a+H

1l m2k0a

=DlH1l m3k0a (12b)

By analogy to a waveguide Maxwellrsquos equations requirethat Ezr minus Erz = iH Since dErdz= 0 and themagnetic fields must be continuous on each side of theinterface the derivative dEzdr must also be continuousacross the interfaces Thus

boundary 1

m1AlJlprimem1k0b= m2BlH

2l

primem2k0b

+H1l

primem2k0b (13a)

boundary 2 m2BlH2l

primem2k0a+H1l

primem2k0a

= m3DlH1l

primem3k0a (13b)





where the prime indicates the derivative of the Bessel func-tion with respect to its argument This notation is fre-quently used but can be confusing so for clarity we canwrite it explicitly

J primel nkx=

dJlmkxxrarrab

mk middotdx (14)

Dividing Eqs 13(b) by 12(b) gives

m3H1l

primem3k0a

m2H1l m3k0a

= BlH2l

primem2k0a+H1l

primem2k0a

BlH2l m2k0a+H

1l m2k0a

(15)

where Bl can be obtained by dividing Eqs 13(a) by 12(a)

Bl=m2Jlm1k0bH

1l

primem2k0bminusm1Jlprimem1k0bH

1l m2k0b

minusm2Jlm1k0bH2l

primem2k0b+m1Jlprimem1k0bH

2l m2k0b

(16)These equations are equivalent to stating that a mod-

ified log-derivative function ur = 1k0dEzdrEzholds continuous across any dielectric boundary for theTE polarization46

We need to find complex values for k0 that are solu-tions of Eq (15) There are in fact an infinite numberof possible solutions corresponding to all the possibleradial mode orders Only the lowest-order modes (eg n=1ndash3) will typically be important since the higher ordersare lossy and will not appear in the WGM spectra Thisrequires Eqs (15) and (16) to be entered into a mathpackage with a complex numerical root solver such asMathematicarsquos ldquoFindRootrdquo function One fixes the desiredangular number l and the various capillary parametersand enters a reasonably good guess for k0 (eg k0 =lmeffa where meff is an estimate of the effective refrac-tive index for the mode The root solver then completes thework

Once the complex root k0 has been found the field (orits radial function) must be plotted in order to find whichradial mode order the solution represents In order to findthe electric field profile one must plot Eq (10) usingthe value found for k0 and the various cylinder parame-ters (radius a film thickness t and the refractive indices)The values of Al Bl and Dl must at this point be set toensure that the field is continuous across the boundariesThe constant Bl can be obtained from Eq (16) once k0has been determined numerically and then Al and Dl aresimply

Al =BlH

2l m2k0b+H

1l m2k0b

Jlm1k0b(17a)

Dl =BlH

2l m2k0a+H

1l m2k0a

H1l m3k0a

(17b)

The field and the various coefficients will contain bothreal and imaginary parts The way to deal with this forcalculating the volume-integrated energy densities (which

must be real) will be discussed later The relative magni-tude of the field is given by the absolute value

Ez =radicReEz

2+ ImEz2 (18)

and the intensity Iz is proportional to Ez2The radial functions for several different TE modes are

shown for an LCORR and an FCM structure in Figure 3from the solutions of Eq (15) We can see some simi-larities and differences between the two structures In theLCORR the outer index is air whereas in the FCM theglass capillary itself is the outermost material (the effectsof the relatively remote outer wall seem to be measurablebut are fairly minor)47 Thus in an LCORR the mode fieldis better confined toward the inner region of the struc-ture which is an advantage for sensing applications In theFCM the smaller index contrast between the fluorescentfilm and the capillary wall effectively ldquoforcesrdquo the fieldfurther away from the channelFor the TM polarization we have the H-field along the

cylinder axis and the radial and azimuthal componentsof the magnetic field are zero The Helmholtz equationis slightly different for this case Assuming that has noazimuthal dependence

2Hzrminus1

r

Hzr

r+k20m

2Hzr= 0 (19)

The second term simplifies to minus2m middot mr middotHzr which can be set to zero since we furtherassume no radial dependence of m1 m2 or m3 within eachof the layers The result is therefore again a set of Besselfunctions this time specifically for Hz and one can write

Fig 3 First order (left) and second order (right) electric field profilesSR for an FCM (top) and an LCORR (bottom) All modes have aresonance wavelength close to 800 nm The LCORR outer radius was50 13m and the wall was 2 13m thick For the FCM the inner radius was50 13m and the fluorescent film was 07 13m thick with a refractive indexof 167 These values are close to typical experimental values for bothstructures The dashed grey lines define the boundaries of the high-indexlayer





a piecewise solution identical to Eq (10) but with differentcoefficients

T R =

⎧⎪⎪⎨⎪⎪⎩

alJlm1k0r r le b

blH2l m2k0r+H


dlH1l m3k0r r gt a

⎫⎪⎪⎬⎪⎪⎭(20)

T R is equivalent to the profile of HzR There can beboth radial and azimuthal parts of the electric field whichcan be written in vector form48

minusrarrE = ErRr +ERe

il (21)

where analogous to Eq (11) we have

ErR=minus 1k0m

2

l

rT mk0r

ER=minus i

k0m2

dT mk0r

dr(22)

EzR = 0

In practice we will only care about the first part ofEq (22) since Er will be what we measure experimentallyas the TM-polarized E-fieldThe boundary requirements in this case are that Hz and

E must be continuous According to Maxwellrsquos equa-tions for the tangential field we have Hzr minus Hrz =im2E Similar to the previous case Hrz = 0 butsince E is continuous the derivative of the magnetic fielddHzdr must change by a factor of m2 on crossing aboundary At the dielectric interfaces we therefore have

boundary 1

1m1alJprimel m1k0b= 1m2blH

2l

primem2k0b

+H1l

primem2k0b (23a)

boundary 2 1m2blH2l

primem2k0a+H1l

primem2k0a

= 1m3dlH1l

primem3k0a (23b)

Thus we find another modified log-derivative func-tion that must be continuous given by vr =1k0m

2HzrHzBy setting the fields and the derivatives (divided by m2)

continuous across the interfaces we obtain

m2H1l

primem3k0a

m3H1l m3k0a

= blH2l

primem2k0a+H1l

primem2k0a

blH2l m2k0a+H

1l m2k0a

(24)

and

bl=m1Jlm1k0bH

1l


1l m2k0b

minusm1Jlm1k0bH2l


2l m2k0b

(25)Equations (24) and (25) are solved in the same way as

(15) and (16)

Equation (20) is often called the transverse magneticldquoradial functionrdquo T r which describes the magneticfield Hz For the TM situation one will usually be mea-suring only Er that is the electric field pointing alongthe radial direction The continuity of the electric displace-ment Er implies that there will be a jump or discon-tinuity in the electric field across each boundary for thispolarization49 as can be seen in Eq (22) Thus once thecoefficients in Eq (20) have been determined the electricfield profile Err has to be adjusted by the ratio of thedielectric constant for each layer as illustrated in Figure 4for Err

2In order to visualize the WGM field around the capil-

laryrsquos cross section we need to consider the azimuthal partof Eq (1) given by by eplusmnil Multiplication of the radialfunction by eplusmnil gives a rotating wave pattern for regionswhere r lt simlk in the region dominated by Jlm1kbHowever as can be seen in Eq (9) the Hankel functionshave an eikr dependence which becomes obvious at large r This results in radial dependence of the phase which com-bined with the azimuthal one forms a spiral pattern in theregions where r gt simlk within which the Hankel func-tions dominate the radiation pattern (Fig 5) The sense ofthe spiral (clockwise or counterclockwise) depends on thesign of eplusmnilFinally the time dependence of the field is included by

replacing eplusmnil with eiplusmnlminusckt In this way ldquomoviesrdquo ofthe WGM field can be produced Since the Hankel func-tion has the asymptotic form eimkr+lminusl2minus4minusckt (egfor counterclockwise rotation) we can see from Eq (9)that increasing t must be balanced by increasing r (forconstant l) in order to obtain a wavefront of constantphase The wave spirals outward in time with the realand imaginary parts phase shifted by 2 These radiatingwaves are the solutions for LCORRs which are pumped

Fig 4 The TM-polarized electric field intensity [Err]2 for a disk

with a= 3 13m b= 27 13m and refractive indices m1 = 133 m2 = 145and m3 = 100 with l = 35 and 0 = 660179 nm There are disconti-nuities in the relative intensity across the boundaries The inset showsthe radial function [T r]2 at the outer boundary (Eq (20)) showing thecontinuity of the function and the change in the slope at the glass-airboundary





Fig 5 Plot of the field amplitude for a counterclockwise-propagatingfirst-order WGM in a layered disk with m1 = 145 m2 = 167 and m3 =133 TE polarization The spiral structure in the outer region is a resultof the Hankel function The non-rotating fields shown in Figure 1 are thesuperposition of clockwise and counterclockwise modes

evanescently via a tapered fiber For FCMs the interfer-ence of the clockwise and counterclockwise-propagatingwaves yields stationary standing wave patterns like thosein Figure 1

One frequently finds Eq (7) cast into a quantum-mechanics-like format with a ldquopotential energyrdquo term thatby analogy can provide additional insights into the prop-erties of the resonances and the mechanisms by whichlight is transferred (or ldquotunnelsrdquo) from the boundary of acylinder or sphere toward the outside region41465051 Forthe case of cylinders the relatively straightforward methodto obtain a Schrodinger-like equation is clearly shown inRef [50] However these formats are not necessary inorder to be able to calculate and plot the WGM fields

4 MODE CHARACTERISTICSThe WGM Q factors can be directly calculated from theequations in the previous sections The real part of k0 givesthe resonance wavelength 0 = 2Rek0 and the modequality factor Q= Rek02Imk0 of the mode Assum-ing no material-related losses then the Q factors are con-trolled only by radiative losses and are mainly governed bythe refractive index contrast between the confining layerand the outer material the thickness of the layer and thediameter of the capillary As shown in Figure 2 the largeindex contrast between the outer capillary wall and outsideair (145 vs 100) as compared to a typical fluorescentlayer and the inner wall (eg sim 16 vs 145) lead to largerQ factors for the LCORR as compared to the fluorescent-type structures For typical LCORR parameters the calcu-lated WGM Q factors can be enormous (eg gt 1015 forthe first-order radial modes of a 100-13m-diameter capil-lary) Since experimental observations give Q = 106ndash107

for structures in this size range52 other effects such as

scattering or absorption must limit the ultimate achievableQ factor For FCMs the calculated Q factors are typicallyin the range of 103ndash106 depending on the film thicknesswhereas experiment gives Q asymp 10353 For both structuresthe higher radial orders are generally lossier and tend tohave lower Q as long as there are no material-relatedlossesMaterial losses can also be modeled by using a com-

plex refractive index in the equations shown in the previ-ous section These losses lower the achievable Q factorwhich could in fact be a sensing transduction mechanismin some cases54 The total Q factor is related to the variousloss parameters according to 1Qtotal = 1Qrad+ 1Qabs+1Qsca where Qrad Qabs and Qsca are the radiationlimited absorption limited and scattering limited Q fac-tors respectively Which loss effect dominates depends onthe size and structure of the resonator for small or highcurvature structures Qrad is usually the limiting parame-ter whereas for large ones material absorption or scatter-ing represent the ultimate limit5556 Figure 6 illustrates theeffect of absorption in the analyte assuming the typicaldevice parameters given in the figure caption47

In an LCORR the free spectral range (FSR the fre-quency spacing between modes) finesse (the ratio of themode spacing to the mode width) and visibility (a measureof mode intensity compared to the background) are rarelymeasured experimentally since the tunable laser measure-ments are typically done over a range much smaller thanone FSR In an FCM data is simultaneously collected overa range of wavelengths much larger than the FSR so theperiodicity in the WGM spectrum should be easily observ-able (eg Fig 7) This can be beneficial for detectingsmall shifts for sensing applications for example by usingFourier methods to measure the resonance shifts over thewhole spectrum rather than using just one peak5758

Fig 6 Effect of absorption or scattering losses in the analyte for anLCORR (left) with a = 50 13m and an FCM (right)47 with a = 15 13mas a function of the confining layer thickness (index m2 = 145 for theLCORR and 167 for the FCM) The outer layer m3 was either air(LCORR) or glass with m3 = 145 (FCM) and the channel index m1 =133 For the LCORR the Q factors are so high in the absence of allbut radiation losses that the = 0 case (no material losses) is not shownThe outer capillary radius of the LCORR was 50 13m the same as theinner capillary radius of the FCM





Fig 7 (a) TM-polarized spectral-spatial mode map along the lengthof the capillary shown in the fluorescence image on the right side ofpanel (a) The variations in the resonance wavelengths are due to changesin the fluorescent silicon quantum dot film thickness along the length ofthe capillary The substructure in the zoomed-in region (red box) is dueto spiraling modes (b) shows representative ldquo1 dimensionalrdquo spectra forwater (blue) and ethanol (red) in the channel Modified From [47] SLane J Chan T Thiessen and A Meldrum Whispering gallery modestructure and refractometric sensitivity of fluorescent capillary-type sen-sors Sensors and Actuators B Chemical 190 752 (2014)

5 SENSITIVITY AND DETECTION LIMITS51 Calculating the Refractometric SensitivityThere are several related ways to calculate the sensitivityto changes in the channel index Perhaps the simplest onestarts by noting that the resonant wavelength is approxi-mately given by

0 asymp2a

lf1m1+ f2m2+ f3m3 (26)

Here fi is simply the fraction of the mode energy in theith layer and the term in the brackets is the average refrac-tive index a is the capillary radius and l is the angularmode number Taking the derivative yields

S = d

dm1

asymp 2af1l

asymp 0f1m2

(27)

where it is assumed that all of the light is in layer 2 so thatlasymp 2m2a0 Obviously this approximation is crude butit does give a reasonable estimate of the sensitivity as wewill show belowAlternatively the sensitivity can be calculated in a more

fundamental way

S = d

dm1

= minus2dkk20

dm1

=minus 0

dm1

middot dkk0

(28)

Here dkk0 is the fractional shift in the magnitude ofthe wavevector induced by a small change in the refrac-tive index m1 Teraoka and Arnold59ndash61 showed for a lay-ered microsphere that this fractional shift depends on thefraction of the mode energy within the analyte They alsoshowed how to estimate the fractional shift using pertur-bation theory

dk

k0=minus1

2

intVpr

minusrarrElowast

0 middotminusrarrEpd3r

intVrminusrarrElowast

0 middotminusrarrE0d

3r(29)

in whichminusrarrE0 and

minusrarrEp represent the initial electric field

and the one perturbed by the index change in the chan-nel respectively The denominator is integrated over allspace (V ) while the numerator reflects the volume Vp inwhich the dielectric constant has been changed by d (theinner layer)We showed how to solve these integrals in Ref [62]

yielding the following equations for the refractometric sen-sitivity of capillary structures

STE = 0

m1

middot I1I1+ I2+ I3

(30a)

STM = 0

2k21

m31k

20 + klowast02

middot(bal2

(dJ lowast

l m1k0r

dr

∥∥∥∥b

Jlm1k0b+ J lowastl m1k0b

times dJlm1k0r

dr

∥∥∥∥b

)+ k20+ klowast0

2

k201I1

)

middot([

I1m2

1

+ I2m2

2

+ I3m2

3

])minus1

(30b)

where the ldquoIrdquo terms represent the volume-integrated elec-tric field energy within layers 1 2 or 3 the ldquolowastrdquo indicatesthe complex conjugate and (k01 k02 are the (real imag-inary) parts of the resonance wavevectorThe energy terms can be obtained by numerically inte-

grating the following expressions

I1 =int m1k0 1b

0xAlJlx2 dx (31a)

I2 =int m2k01a

m2k0 1bxBlH

2l x+H

1l x2 dx (31b)

I3 =int

m3k0axDlH

1l x2 dx (31c)

where x equiv x1+ ik02k01 and x equiv k01nr Equation (31)is written explicitly for the TE case for the TM polar-ization one replaces the Al Bl Dl coefficients with theTM ones Equation (31) can require some care in settingthe integration limit since in practice Eq (31c) divergesalthough the divergence might not always be apparent upto an integration limit close to Qa For low Q-factors thefield may never approach zero outside the outer boundarywhich will lead to further issues in setting the integrationlimitFor biosensing applications the refractive index changes

will be localized to the region immediately at the capillaryinner wall In that case one does not have a uniform changein the refractive index as assumed in the previous equa-tions and the sensitivity does not have the same meaning(since there is effectively no dm1) Teraoka and Arnoldshowed how the resonance shift related to a uniform layerof adsorbed particles can be calculated6061 Equation (30)





becomes slightly modified since one instead has to calcu-late the shift due to a uniform density of small particlesof index np coated on the channel surface For a uniformrefractive shift Zhu et al63 showed that the fractional shiftdue to biomolecular absorption can be estimated as

d

= pex

2radicm2

2minusm21

020

m2

m21

S (32)

where 0 is the vacuum permittivity p is the biomoleculesurface density ex is the biomoleculersquos excess polariz-ability and S is the sensitivity calculated previously Thefractional shift depends linearly on the device sensitivityand the concentration of surface particles

52 Detection Limits Getting the MostOut of the Measurement

The detection limit (DL) is the most important figure ofmerit for microfluidic capillary sensors The DL dependson the sensitivity of the device and the wavelength shiftresolution limit min of the measurement system and isgiven by

DL= minS (33)

in dimensionless units A low detection limit requires asmall shift resolution limit and a large sensitivity Twodifferent methods can be used to sense the resonanceshift6465 In the ldquointensity-shiftrdquo method the transmissionis measured through a waveguide or fiber taper evanes-cently coupled to the device at a fixed wavelength Thelaser should be tuned to the maximum slope of the WGMand the shifts are measured as intensity changes Thismethod can be very sensitive but it has a small dynamicrange for changes in the capillary channel index In thewavelength-shift method a tunable laser is scanned acrossa resonance and the peak shift is measured

In one approximation the wavelength shift resolutionwas considered inversely proportional to the Q factor andsensitivity of the WGM such that DL QSminus166 Unfor-tunately this leads to some inconsistency in the literaturewith different works assuming various wavelength shiftresolutions as fractions of the resonance linewidth67 Sev-eral subsequent works have assumed that the detectionlimit is inversely related to the Q factor However theseassumptions are too simplified good detection limits canalso be achieved with low Q factors depending on the datasampling noise levels and data analysis methods5768

If the peak wavelengths are to be found via Lorentziancurve fitting Silverstone et al57 used numerical modelingto find min over a fairly wide range of experimentalconditions within a 3 level of certainty

min3=22 middot10minus20 middotf 029 middotP 065

SNR051meters (34)

Here f is the width of the WGM in units of Hz (whichcan be directly converted from wavelength via f =

c2) P is the sampling rate also in Hz (ie spectralfrequency range per pixel on the CCD) and the SNR ison a linear scale The Q factor does play a role via f butit is a rather weak effect (ie changing the linewidth bythree orders of magnitude only changes the 3 resolutionmin by a factor of sim75) holding everything else con-stant and assuming that the sampling rate is greater thanthe Nyquist rateFor an FCM the spectrometer will typically collect data

over a range of wavelengths much larger than the FSRSilverstone et al found that this case lends itself naturallyto Fourier analysis techniques Accordingly they appliedthe Fourier shift theorem over the periodic WGM spectrumto find the wavelength shifts induced by different analytesin the channel A numerical analysis over a wide rangeof conditions showed that the detection limit is only veryweakly (and inversely) dependent on the linewidth57

min3=175 middot10minus14P 050

f 012 middotSNR052meters (35)

Here increasing the Q factor (ie decreasing the WGMlinewidth) by a factor of 1000 worsens the wavelengthresolution by a factor of about 23 as long as the linewidthremains smaller than the FSR Essentially higher Q leadsto poorer sampling of each peak and ultimately decreasesthe ability to resolve spectral shifts

6 EXPERIMENTAL TECHNIQUES(FABRICATION FUNCTIONALIZATIONEXCITATION AND ANALYSIS)

Experimental details for making LCORR and FCM sensormeasurements are rarely included in standard publicationswhere only a short experimental section is typically givenFortunately two experimental papers have been publishedthat focused on the laboratory aspects of LCORRs69 andFCMs70 Essentially the experimental aspects can be bro-ken down into three parts

61 Sample FabricationMaking an LCORR or FCM is non-trivial LCORRs areusually synthesized by a combination of heating andpulling and etching with hydrofluoric acid The heatingand pulling technique can be adapted from custom-builtfiber tapering stations These often use a hydrogen torchor CO2 laser to locally heat the sample which is simulta-neously stretched by attaching the sample to small servomotors on each side Typically a bank of ldquorecipesrdquo for thetemperature time and pulling rate has to be custom builtto produce the desired structure which takes some amountof trial and errorA problem with the LCORR is that there will be uncer-

tainty in the capillary wall thickness unless the recipes arewell characterized and reproducible Since the overall sen-sor performance is highly sensitive to the wall thickness





further thinning to a desired value is often done by pump-ing HF through the channel while monitoring the opti-cal response of the capillary This obviously requires theuse of dangerous chemicals and is one of the drawbacksto LCORR fabrication However an advantage is that thewall thickness can be controlled quite precisely The finalcapillary is extremely fragile and a considerable amountof care is required in handling itWith FCMs the biggest fabrication difficulty is making

a uniform luminescent film of the right thickness inside thecapillary Both silica-embedded quantum-dot71 and dye-doped polymer films58 have been synthesized McFarlaneet al70 reported that the success rate is quite low (on theorder of a few percent) and is sensitive to the age of thechemicals due to oxidation and hydration issues The filmthickness is difficult to control although finer tuning couldlikely be obtained for example by etching the layer con-taining the QDs Rolled ldquotuberdquo FCMs are another alter-native72 but these are difficult to interface to a pumpingsystem

62 SetupOnce the sample has been fabricated the next issue isgetting it hooked up to a syringe pump and associated tub-ing system Norland-type adhesive (eg 76) yields goodresults on polyethylene tubing and standard solvents suchas methanol ethanol water or phosphate buffer solutionsbut it appears at least partly dissolved by acetone Typi-cally tests need to be conducted to ensure that the adhesiveis compatible with the tubing and solvent used otherwisethe seal between the tubing and the capillary can weakenand eventually begin to leak The tubing length shouldbe minimized in order to avoid the experiment taking along time as the solvents are slowly pumped through longtubing lengths A pumping rate of a few microLitres perminute or less is typical using a standard syringe pump

63 Data CollectionFor an FCM the only requirement is a system to col-lect and analyze the luminescence A microscope objec-tive coupled to a simple spectroscopy system such as theSGS from the Santa Barbara Instruments Group (now nolonger produced) or the LHires from Shelyak work wellThe WGM spectrum can be collected over a wide rangeof wavelengths much larger than the free spectral rangeThus many modes are collected at the same time althoughat the cost of a relatively low signal For an LCORRa tapered fiber must be fabricated and brought near thecapillary wall using piezo-controllers Then light from atunable laser is coupled into the taper and scanned acrossone or more WGM resonances Alternatively the LCORRcan be interfaced to a system of optical waveguides ona chip73 The WGMs appear as dips in the transmissionspectrum collected by a photodiode at the opposite end ofthe taper The signal-to-noise ratio is much higher for a

given collection time than for the fluorescence method butthe wavelength range that can be collected is narrowerEach technique has a specific set of advantages and dis-

advantages The LCORR is considerably more difficult andexpensive to analyze but it yields better detection limits ascompared to the FCM which in turn is far more durableand robust

7 PITFALLS71 Thermal EffectsOne of the biggest issues with capillary-based WGM sen-sors is thermal drift Temperature fluctuations in the lab-oratory or as a result of absorption of the pump lightmodify the refractive index of all three layers via thethermo-optic effect Thermal expansion can also affect theresonant wavelength For example in an FCM the reso-nant wavelength was found to shift by around 1 pmmWof laser power used to excite the fluorescent layer71 Con-sidering that shifts as small as 1 pm can theoreticallybe detected via Fourier methods temperature fluctuationsmay represent the ultimate barrier toward achieving lowerdetection limits In LCORRs laboratory temperature driftsas small as 1ndash2 K can be significant74

The magnitude of the temperature-induced wavelengthshift can be obtained by noting that a change in theeffective index or capillary radius produces a propor-tional change in the resonant wavelength which can beassumed linear Thus d0 = dneffneff+daa If thesechanges are temperature-dependent we can divide bothsides by dT yielding

d

dT= 0

(1

meff

dmeff

dT+ 1

a

da

dT

)(36)

The second term on the right side in the brackets isthe linear thermal expansion coefficient of the capillary and meff is approximated by f1m1+ f2m2+ f3m3 withfi being the energy fraction in the ith layer Using theexpressions derived above for the fractional intensities thetemperature-induced wavelength shifts for both polariza-tions isdmin TM

dT= 0

(I1m

211+ I2m

222+ I3m

233

I1m1+ I2m2+ I3m3

+

)

(37)

dmin TE

dT= 0

(I11+ I22+ I33

m1I1+m2I2+m3I3+

)(38)

where the thermo-optic coefficient in the ith layer is i =ddTi Finally the temperature-shift-induced detectionlimit can be found by multiplying Eq (33) by dT dT

DLthermalTETM =dmin TETM

dTmiddot dT

STETM(39)

The term dTS is the ratio of the thermal fluctuationin the experiment to the sensitivity of the sensor Equa-tion (39) is the refractometric detection limit due to tem-perature fluctuations alone The overall detection limit is





then the sum of the resolution-limited one (Eq (33)) andtemperature-limited one (Eq (39))

Figure 8 shows the temperature-fluctuation-based detec-tion limits for an FCM and an LCORR assuming alaboratory fluctuation of 1 K For different temperaturefluctuations the results scale proportionally For an FCMa zero thermal detection limit can be achieved for a suffi-ciently thin film (eg sim400 nm as in Fig 8) but for moretypical thicker films a low DL requires either a large neg-ative thermo-optic coefficient or a large refractive index ofthe solvent This is because only a relatively small frac-tion of the mode energy extends into the channel Foran LCORR a wall thickness of sim13 13m can produce azero thermal DL within the range of m1 and of typical

Fig 8 Contour maps of the thermal detection limit for a typicalFCM (top) and LCORR (bottom) as a function of the refractive index(abscissa) and thermo-optic coefficient (ordinate) of the analyte FCMfirst radial order TE polarization l= 190 a= 15 13m fluorescent layerthickness t = 04 13m and m2 = 167 LCORR first radial order TEpolarization l= 550 a= 50 13m wall thickness t = 13 13m m2 = 145Both structures can theoretically achieve a zero thermal limit (inside thedark red band) under normal conditions if the wall thickness can be con-trolled precisely

solvents such as water for the first order mode The opti-mal wall thickness depends on the radial mode order to bemeasuredTwo papers recently showed that there can be an opti-

mum thickness at which the thermo-optic coefficient ofthe silica-based layer 2 can be balanced by the nega-tive coefficient of water leading to a zero net thermalshift7475 For this specific thickness the thermal detectionlimit (Eq (39)) becomes zero These zero drift ldquovalleysrdquoare within the dark red bands for the two specific exam-ples shown in Figure 8 In the case of an FCM gettinga DLthermal = 0 depends on having a very thin fluorescentcoating In many practical cases the thermal drifts canrepresent the ultimate detection limit of the device

72 Mechanical DriftA second aspect that can limit the achievable DL ismechanical drift In the case of an LCORR mechanicalinstabilities can lead to relative motion between the fibertaper and the capillary Since the taper must be held veryclose to the LCORR in order to have efficient coupling thetaper drift can affect the Q factor and resonant wavelengthof the device In the simplest approximation the gap-dependence of the linewidth has been approximated as76

= 0

Q0

(1+ e

0

)(40)

Here Q0 is the ldquointrinsicrdquo Q factor (in the absenceof the taper) e is the taper-to-disk coupling rate and0 is the cavityrsquos intrinsic loss rate which is simply2c0Q0 The value of e depends exponentially onthe gap e = e0e

minusg where e(0) is the coupling ratewhen the cavity and the taper are in contact describesthe strength of the coupling and g is the gap distanceThese values need to be obtained experimentally for GaAsnanodisks with a silica fiber taper was found to be00025 nmminus1 and e(0) asymp 15 GHz Typically the linewidthmay change by a few hundred pm upon changing thegap distance from 350 to 200 nm at an increasing rateas the gap narrows77 In the case of intensity-shift sensortransduction the transmission at a fixed wavelength alsodepends on the taper-cavity distance via78

T =(1minuse0

1+e0

)2

(41)

In an LCORR taper-capillary drift can cause smallwavelength shifts While the magnitude of such shifts as afunction of gap cannot be readily calculated redshifts upto 8 nm have been reported for low-angular-order WGMsof silicon microdisks79 while other theoretical studieshave suggested a more complicated gap dependence80 InLCORRs the drift-induced wavelength shifts are probablyquite small and likely do not appreciably affect the detec-tion limits as much as thermal drifts or the wavelengthshift resolution limits





For FCMs the main issue is mechanical drift withrespect to the entrance slit of the spectrometer Whilethis effect cannot easily be quantified we have found thatmechanical motion of the capillary or the microscope stageitself can over a period of hours significantly affect theresults If the capillary axis is aligned perpendicular to theentrance slit any drift perpendicular to the capillary axis(ie along the length of the slit) will change the verticalposition of the spectral image on the CCD array This canlead to data processing and wavelength calibration issuesParallel drift (along the capillary channel) means that aslightly different part of the capillary is exposed in eachimage Unless the capillary parameters (ie film thicknessand inner diameter) are uniform along the length of thedrift artificial shifts will appear in the sensorgrams

73 Spiral ModesA third effect (after thermal fluctuations and drift) is moreclosely related to the physics of capillary WGMs All ofthe calculations in the previous section assumed that therewas no z-component of the electric field (ie kz = 0 alongthe capillary axis) However both LCORRs and FCMs canshow strong spiraling mode effects81 Spiral or ldquoskewrdquomodes are well known in optical fibers82 and have beendescribed with respect to cylindrical WGMs84 For smallspiraling angle (Fig 9) the resonance wavelength canbe estimated as82

asymp a

l

(2meffminus

2

meff

)(42)

Equation (42) shows that spiral modes should appear onthe short-wavelength side of the main WGMSpiral modes can appear as a set of peaks that ride

on the short-wavelength tail of the main WGMs Whetherthese peaks are resolvable or simply appear as a short-wavelength skewing of the main WGM seems to depend

Fig 9 Diagram showing ray path of a spiral WGM (blue line) Theangle is used to calculate the associated resonance wavelength usingEq (42)

on the experimental conditions such as the thickness ofthe confining layer83 A key aspect has to do with angulardistribution of the illumination In the case of LCORRsZamora et al84 showed that the excitation of the spiralmodes could be controlled to some degree by the taperdiameter Wider tapers produce a narrow angular distribu-tion of the excitation light and were found to reduce theappearance of spiral modes The drawback is of coursethat the coupling rate decreases due to the smaller evanes-cent extension of the field for a thicker taper In the caseof FCMs the luminescence is mainly isotropic (to a firstapproximation ignoring Purcell effects) regardless of theexcitation condition and therefore spiral mode skewing ofthe main WGMs is ubiquitous and unavoidableThe presence of spiral modes can have minor detrimen-

tal effects on the sensing performance since it essentiallybroadens and skews the WGMs If the WGMs becomeskewed they cannot be modeled using a Lorentzian line-shape Thus if one wants to use curve fitting methods tomeasure peak shifts one needs a function that can ade-quately describe the skewed mode shape Manchee et al71

showed that the following ldquoskewed Lorentzianrdquo85 equationfits the WGMs quite well although it lacks physical basiswith respect to WGMs

Pf = 2Af

[1+4

(f minus f0f

)2](43)

Here f0 is the central frequency A is a normalizingfactor and the peak width describes the frequency-dependence of the skewing according to

f = 1+B0

1+B middot ecfminusf0(44)

where c is another fitting parameter

8 EXAMPLES OF RELEVANT SENSORSSeveral potential sensor applications have been exploredand experimentally verified Most of the work has focusedon LCORRs since these structures have been studied fora longer time and by more research groups This is notmeant as an exhaustive sampling of all such work butsimply as a brief overview to illustrate the almost limitlesspossibilities of refractometric capillary-based biosensingSome studies show physical versatility of these systemswhich are able to handle almost any solvent includingviscous dense oils86 and in many cases true chemicalbiosensing has been demonstratedThe most common channel wall functionalization

scheme so far is based on the standard silanization reac-tion8788 This starts with a clean ndashOH-terminated surfacewhich can be obtained by HCl washing followed by rins-ing with methanol water or other alcohols or by anneal-ing in a hydrogen- or moisture-rich ambient or otherpreparation methods89 The next step is typically to pump





an amine-terminated silane solution (eg aminopropyl-tri(m)ethoxysilane APT(M)S or APT(E)S) into the capil-lary The CH3 end of the APTS molecule hydrolizes withwater forming an OH termination which then binds withndashOH terminated silica surface to bind the silane throughan SindashOndashSi link producing another water molecule in theprocess (Fig 10)

This reaction is sensitive to the water concentration90

even 1ndash2 water in the APTS in (m)ethanol solutionscan cause extensive polymerization of the APTS and theformation of an uneven and non-uniform functionalizedsurface Such surfaces tend to be unstable and susceptibleto the subsequent undesired removal of ldquochunksrdquo of poorly-bound polymerized material Anhydrous functionalizationcan be desirable if enough water to maintain the reactionis available from the surface reaction Ideally the resultingsurface would consist of a monolayer of APTS on the chan-nel wall with the ndashNH2 terminus available for reaction butthis can be difficult to achieve on a consistent basis91

Some of the earliest examples with LCORRs involvedsingle-strand DNA sensing9293 Single-stranded (or double

Fig 10 Silanization scheme illustrating surface silanization (hydrolysisand condensation steps) followed by binding of biotin to the silanizedsurface Modified From [101] N A Lapin and Y J Chabal Infraredcharacterization of biotinylated silicon oxide surfaces surface stabilityand specific attachment of streptavidin J Phys Chem B 113 8776(2009) copy 2009 From [91] J A Howarter and J P Youngblood Opti-mization of silica silanization by 3-aminopropyltriethoxysilane Langmuir22 11142 (2006) copy 2006 From [102] S McFarlane Biosensing usingfluorescent-core microcapillaries University of Alberta Masters Thesis(2012) copy (2012)

stranded)94 DNA fragments can bind to the amine-terminated silanized surface layer Detection limits werefound to be on the order of 4 pg per square millimeter ofthe channel surface corresponding to a density of about1010 moleculescm2 for an LCORR with a sensitivity ofabout 37 nmRIU which is high for a first-order WGMThe same group later demonstrated the binding of double-stranded DNA to methyl binding proteins95 rather thanusing the silane-binding schemes discussed in the previousparagraphBiotin-avidin reactions are one of the most common test

systems for new biosensor technologies mainly becausethe binding chemistry is well understood and because ofthe exceptionally strong chemical affinity between biotinand streptavidin96ndash98 The functionalization can be basedon the silane chemistry described above since the NH2

ends of the silane molecules bind to biotin through its car-boxyl end Alternatively one can attach biotinylated BSA(bovine serum albumen) to the ndashNH2 endgroups of thesilane molecules Using biotinylated BSA has the advan-tage that the expected wavelength shift is larger than itwould be for the much smaller biotin molecule BSA is alarge protein that can have both carboxyl and amine endgroups and can thus bind to biotin and to the silanizedcapillary channel Non-biotinylated BSA can also be usedto block the silanized surface against nonspecific stickingof streptavidin or other proteins99 This can be a usefulcheck that the surface chemistry is occurring as expectedA third option is to use silane functionalized polyethyleneglycol (PEG) with the desired endgroup (eg silane-PEG-biotin)100

The last step is then the reaction of (strep)tavidinwith the surface-bound biotin The streptavidin proteinis large compared to biotin essentially the tetrahydroth-iophene end of the biotin fits almost perfectly into theldquodivotsrdquo in the streptavidin in one of the strongestnon-covalent biomolecular binding reactions103 Success-ful tests have been performed both for microspheres104

which was one of the earliest demonstrations of capil-lary WGM based biosensing and FCMs105 although inthe latter case a significant amount of nonspecific strep-tavidin binding was reported The reverse binding inwhich streptavidin was attached to a surface that was firstsilanized and then activated for streptavidin using a bis(sulfosuccinimidyl) suberate solution was used to showsmall molecule (biotin) detection in a highly sensitiveLCORR106 Direct BSA binding to a silanized LCORR hasalso been established63

Several other types of biomolecular sensing havebeen demonstrated mainly also involving some form ofsilanization surface chemistry For example T4 bacterio-phage virus was detected in an LCORR at a level of23times 103 pfumL (plaque-forming units) using silaniza-tion followed by glutaraldehyde cross linking107 TheR5C2 phage was also detected in an LCORR usingthe same surface functionalization method and detected





via streptavidin binding after blocking with BSA108

The 6-benzylaminopurine plant growth regulator wasdetected at a level of 001 mgkg using the same initialsilanization+glutaraldehyde steps followed by the appli-cation benzylaminopurine riboside antibody to the gluter-aldehyde cross linker109 Phosphate-based pesticides weredetected using similar methods with the addition of AChEenzyme to the glutaraldehyde linker110 Detection limitswere around 38times 10minus11 M HER2 breast-cancer-relatedantibodies were also reported at levels down to 13 ngmLby cross linking the LCORR surface with dimethyl pime-limidate dihydrochloride rather than glutaraldehyde111

Almost all of these methods are based on initial silaniza-tion of the capillary channel This is the key function-alization step because of the potential difficulty withpolymerization and the formation of less stable surfacesit is imperative to run blanks (eg perform the same exper-iment without the silanization step or use a blocking agent)to ensure that the surface binding is indeed specific to thedetected compound The evidently wide range of possi-ble biochemical sensor devices and the obvious success ofmany investigations over the past several years suggest thatLCORRs and potentially FCMs can be versatile and verysensitive and could even compete with surface-plasmon-based methods for many applications The biggest dif-ficulty at the moment is probably associated with theextreme mechanical fragility of thinned capillaries and thenecessity to perform surface chemistry inside these fragilestructures

9 CONCLUSIONS AND OUTLOOKThe cylindrical whispering gallery modes of microcapil-laries represent a highly sensitive and potentially promis-ing means for microfluidic sensing and analysis Thesedevices are known alternatively as liquid core optical ringresonators opto-fluidic ring resonators or fluorescent coremicrocapillaries In all cases they utilize a thin high-refractive-index layer to confine radiation and develop theresonant modes of the structure In LCORRs (which areessentially synonymous with opto-fluidic ring resonatorsor ldquoOFRRsrdquo) the resonances form in the glass walls ofa mechanically thinned capillary and are probed with atunable laser system which is evanescently coupled to thedevice In the FCM case a high-index fluorescent layeris formed on the capillary channel and the resonances areprobed with a spectrometerIn this short review the basic theory that permits the cal-

culation of the mode field profile was first outlined This isthe first necessary step in the study of capillary (or fiber)WGM-based structures LCORR and FCM structures werethen compared and contrasted showing the advantages anddisadvantages of each sensor type The LCORR is fun-damentally a more sensitive device in almost all aspectsmainly because the resonances are better confined dueto the large index contrast between the thinned capillary

wall and the outside air (this contrast is much smallerin an FCM where the outermost effective boundary isbetween the fluorescent layer and the thick-walled capil-lary) However LCORRs have several troubling aspectsincluding mechanical fragility and a rather difficult setupthat requires careful positioning of optical elements FCMsare far more robust but they lack the amazing sensitivityof LCORR-type devicesNext we examined the various experimental setups

that can be used for each structure It can be quitedifficult to determine the experimental difficulties fromthe short ldquoexperimentalrdquo section of most papers hencetwo experimentally-oriented works were cited extensivelyThese efforts can be useful to prevent future investigatorsfrom spending a lot of time overcoming the same prob-lems Finally a short summary of the recent applicationsof these devices was provided with a few key aspectsconcerning the surface chemistry typically required to beperformed in the capillary channelThe future of this field remains open A key aspect for

LCORRs will concern the difficulty of integrating the frag-ile structure onto a workable device In terms of sensorapplications FCMs lag behind LCORRs partly becauseof their lower sensitivity but also probably because onlya few groups have to date been able to make and test suchstructures Alternative cylindrical WGM sensor structuresnot explicitly discussed in this brief review (eg litho-graphically patterned microdisks) are also very promis-ing with at least one company now selling diagnosticdevices based on advanced microdisk technology112 Thebasic cylindrical resonances are mainly identical to thosediscussed here but may include reflections from the topand bottom surfacesFuture work will probably aim towards specific appli-

cations and packaging (ie how to integrate a completedetection system) Simple and more effective synthesismethods for FCMs are also desirable and if the layerindex can be increased the sensitivity and detection limitsof FCMs could approach LCORR values Finally the sil-ica surface chemistry that occurs in the capillary channelsis not always the same as for flat interfaces and has notbeen well-studied The effects of pH can be critical andthe exact values used in the capillary devices are rarelyreported Thus the functionalization chemistry remainsone of the most difficult and hard-to-reproduce steps inthe development of targeted WGM-based capillary sensordevices

Acknowledgments Dr Yanyan Zhi is thanked for aneditorial review

References and Notes1 B H Weigl and O S Wolfbeis Capillary optical sensors Anal

Chem 66 3323 (1994)2 O S Wolfbeis Capillary waveguide sensors Trac Trend Anal

Chem 15 225 (1996)





3 F Baldini and A Giannetti Optical chemical and biochemical sen-sors New trends Proc SPIE 5826 485 (2005)

4 P V Lambeck Integrated optical sensors for the chemical domainMeas Sci Technol 17 R93 (2006)

5 M Borecki M L Korwin-Pawlowski M Beblowska J Szmidtand A Jakubowski Optoelectronic capillary sensors in microfluidicand point-of-care instrumentation Sensors 10 3771 (2010)

6 J Zhang J Hoogmartens and A Van Schepdael Advances incapillary electrophoretically mediated microanalysis An updateElectrophoresis 29 56 (2008)

7 E Dabek-Zlotorzynska V Celo and M M Yassine Recentadvances in CE and CEC of pollutants Electrophoresis 29 310(2008)

8 Z El Rassi Electrophoretic and electrochromatographic separa-tion of proteins in capillaries An update covering 2007ndash2009Electrophoresis 31 174 (2010)

9 M C Breadmore J R E Thabano M Dawod A A KazarianJ P Quirino and R M Guijt Recent advances in enhancing thesensitivity of electrophoresis and electrochromatography in capil-laries and microchips (2006ndash2008) Electrophoresis 30 230 (2009)

10 A P Lewis A Cranny N R Harris N G Green J A WhartonR J K Wood and K R Stokes Review on the development oftruly portable and in-situ capillary electrophoresis systems MeasSci Technol 24 042001 (2013)

11 M E Bosch A J Ruiz Saacutenchez F S Rojas and C B OjedaRecent development in optical fiber biosensors Sensors 7 797(2007)

12 S Calixto M Rosete-Aguilar D Monzon-Hernandez and V PMinkovich Capillary refractometer integrated in a microfluidicconfiguration Appl Opt 47 843 (2008)

13 L E Helseth Simultaneous measurements of absorption spectrumand refractive index in a microfluidic system Opt Express 20 4653(2012)

14 T M Squires and S R Quake Microfluidics Fluid physics at thenanoliter scale Reviews in Modern Physics 77 977 (2005)

15 C Yi C-W Li S Ji and M Yang Microfluidics technology formanipulation and analysis of biological cells Analytical ChimicaActa 560 1 (2006)

16 B Kuswandia Nuriman J Huskens and W Verboom Opticalsensing systems for microfluidic devices A review AnalyticalChimica Acta 601 141 (2007)

17 A Chiasera Y Dumeige P Feron M Ferrari Y Jestin G NConti S Pelli S Soria and G C Righini Spherical whispering-gallery-mode microresonators Laser Photon Rev 4 457 (2010)

18 F Vollmer and S Arnold Whispering-gallery-mode biosensingLabel-free detection down to single molecules Nat Methods 5 591(2008)

19 V Lefeacutevre-Seguin Whispering-gallery mode lasers with doped sil-ica microspheres Opt Mater 11 153 (1999)

20 K J Vahala Optical microcavities Nature 424 839 (2003)21 J-M Geacuterard and B Gayral Semiconductor microcavities quantum

boxes and the Purcell effect Lecture Notes in Physics 531 331(1999)

22 H Yokoyama Y Nambu and T Kawakami Controlling sponta-neous emission and optical microcavities Confined Electrons andPhotons NATO ASI Series 340 427 (1995)

23 M V Artemyev U Woggon R Wannemacher H Jaschinski andW Langbein Light trapped in a photonic Dot Microspheres act asa cavity for quantum dot emission Nano Lett 1 309 (2001)

24 S Arnold Spontaneous emission within a photonic atom Radia-tive decay rates and spectroscopy of levitated microspheres Spec-troscopy of Systems with Spatially Confined Structures NATOScience Series 90 465 (2002)

25 K Vahala Optical microcavities Nature 424 839 (2003)26 A Meldrum P Bianucci and F Marsiglio Modification of ensem-

ble emission rates and luminescence spectra for inhomogeneously

broadened distributions of quantum dots coupled to optical micro-cavities Optics Express 18 10230 (2010)

27 G C Righini Y Dumeige P Feron M Ferrari G N ContiD Ristic and S Soria Whispering gallery mode microresonatorsFundamentals and applications Rivista del Nuovo Cimento 34 435(2011)

28 J B Cole N Okada and S Banerjee Advances in finite-differencetime-domain calculation methods Light Scattering Reviews editedby A A Kokhanovsky Springer-Verlag (2012) pp 115ndash179

29 V Zamora A Diacuteez M V Andreacutes and B Gimeno Cylindri-cal optical microcavities Basic properties and sensor applicationsPhotonics and Nanostructures-Fundamentals and Applications9 149 (2011)

30 Lord Rayleigh The problem of the whispering gallery Philos Mag20 1001 (1910)

31 C V Raman On whispering galleries Proc Indian Ass Cult Sci7 159 (1921ndash1922)

32 G Mie Beitrage zur optik truber medien Ann d Physik 25 377(1908)

33 J J Thomson Notes on Recent Research on Electricity and Mag-netism Intended as a Sequel to Professor ClerkndashMaxwellrsquos Treatiseon Electricity and Magnetism Clarendon Press Oxford (1893)

34 A Clebsch Ueber die reflexion an einer Kugelflaumlche Journal fuumlrMathematik 61 195 (1863)

35 L Lorenz Lysbevaegelsen i og uden for en af plane lysbol-ger belyst kugle Det Kongelige Danske Videnskabernes SelskabsSkrifter 6 1 (1890)

36 L Lorenz Sur la lumiegravere reacutefleacutechie et reacutefracteacutee par une sphegravere (sur-face) transparente In Oeuvres scientifiques de L Lorenz revueset annoteacutees par H Valentiner Tome Premier Libraire Lehmann ampStage Copenhague (1898) pp 403ndash529

37 P Debye Der Lichtdruck auf Kugeln von beliebigem MaterialAnnalen der Physik Vierte Folge Band 30 No 1 57 (1909)

38 H Horvath Gustav Mie and the scattering and absorption of lightby particles Historic developments and basicsJournal of Quanti-tative Spectroscopy and Radiative Transfer 110 787 (2009)

39 T Wriedt Mie Theory A review The Mie Theory editedby W Hergert and T Wriedt Springer Series in Optical Sci-ences Springer-Verlag Berlin Heidelberg (2012) Vol 169 DOI101007978-3-642-28738-1_2

40 A N Oraevsky Whispering-gallery waves Quantum Electronics32 377 (2002)

41 A B Matsko and V S Ilchenko Optical resonators withwhispering-gallery modesmdashpart I Basics IEEE Journal ofSelected Topics in Quantum Electronic 12 3 (2006)

42 L Prkna J Ctyroky and M Hubalek Ring microresonator asa photonic structure with complex eigenfrequency Optical andQuantum Electronics 36 259 (2004)

43 Y V Prokopenko Y F Filippov and I A Shipilova Distribution ofthe field of whispering-gallery modes In a radially nonuniform two-layer cylindrical dielectric resonator Radiophysics and QuantumElectronics 51 561 (2008)

44 C F Bohren and D R Huffman Absorption and Scattering ofLight by Small Particles Wiley (1998)

45 P Bianucci J R Rodriguez F Lenz C M Clement J G CVeinot and A Meldrum Silicon nanocrystal luminescence cou-pled to whispering gallery modes in optical fibers J Appl Phys105 023108 (2009)

46 B R Johnson Theory of morphology-dependent resonances Shaperesonances and width formulas Journal of the Optical Society ofAmerica A 10 343 (1993)

47 S Lane J Chan T Thiessen and A Meldrum Whisperinggallery mode structure and refractometric sensitivity of fluorescentcapillary-type sensors Sensors and Actuators B Chemical 190 752(2014)

48 J A Stratton Electromagnetic Theory McGraw-Hill New York(1941)





49 J R Buck and H J Kimble Optimal sizes of dielectric micro-sphere for cavity QED with strong coupling Physical Review A67 033806 (2003)

50 R Yang A Yun Y Zhang and X Pu Quantum theory of whis-pering gallery modes in a cylindrical optical microcavity Optik122 900 (2011)

51 M M Popov On the problem of whispering gallery waves in aneighborhood of a simple zero of the effective curvature of theboundary Journal of Soviet Mathematics 11 791 (1979)

52 I M White H Oveys and X Fan Liquidndashcore optical ring-resonator sensors Opt Lett 31 1319 (2006)

53 P Bianucci Y Y Zhi F Marsiglio J Silverstone and A MeldrumMicrocavity effects in ensembles of silicon quantum dots coupledto high-Q resonators Physica Status Solidi A 208 639 (2011)

54 T Ling and L J Guo Analysis of the sensing properties of silicamicrotube resonators Journal of the Optical Society of America B26 471 (2009)

55 R D Kekatpure and M L Brongersma Fundamental photophysicsand optical loss processes in Si-nanocrystal-doped microdisk res-onators Physical Review A 78 023829 (2008)

56 L H Agha J E Sharping M A Foster and A L Gaeta Opti-mal sizes of silica microspheres for linear and nonlinear opticalinteractions Applied Physics B 83 303 (2006)

57 J W Silverstone S McFarlane C P K Manchee andA Meldrum Ultimate resolution for refractometric sensing withwhispering gallery mode microcavities Opt Express 20 8284(2012)

58 K J Rowland A Francois P Hoffmann and T M Monro Flu-orescent polymer coated capillaries as optofluidic refractometricsensors Opt Express 21 11492 (2013)

59 I Teraoka S Arnold and F Vollmer Perturbation approach toresonance shifts of whispering-gallery modes in a dielectric micro-sphere as a probe of a surrounding medium Journal of the OpticalSociety of America B 20 1937 (2003)

60 I Teraoka and S Arnold Enhancing sensitivity of a whisperinggallery mode microsphere sensor by a high-refractive index surfacelayer Journal of the Optical Society of America B 23 1434 (2006)

61 I Teraoka S Arnold and F Vollmer Whispering-gallery modesin a microsphere coated with a high-refractive index layerPolarization-dependent sensitivity enhancement of the resonance-shift sensor and TE-TM resonance matching Journal of the OpticalSociety of America B 24 657 (2007)

62 S Lane F Marsiglio Y Zhi and A Meldrum Appl Opt(in review)

63 H Zhu I M White J D Suter P S Dale and X Fan Analysisof biomolecule detection with optofluidic ring resonator sensorsOpt Express 15 9139 (2007)

64 C-Y Chao and L J Guo Design and optimization of microringresonators in biochemical sensing applications Journal of Light-wave Technology 24 1395 (2006)

65 C-Y Chao W Fung and L J Guo Polymer microring resonatorsfor biochemical sensing applications IEEE Journal of SelectedTopics in Quantum Electronics 12 134 (2006)



68 I M White and X Fan On the performance quantification of res-onant refractive index sensors Opt Express 16 1020 (2008)

69 I M White H Zhu J D Suter X Fan and M Zourob Label-freedetection with the liquid core optical ring resonator sensing plat-form Methods in Molecular Biology Biosensors and Biodetectionedited by A Rasooly and K E Herold Humana Press a part ofSpringer Science+Business Media LLC (2009) Vol 503

70 S McFarlane C P K Manchee J W Silverstone J G CVeinot and A Meldrum Synthesis and operation of fluorescent-core microcavities for refractometric sensing Journal of VisualizedExperiments 73 e50256 (2013)

71 C P K Manchee V Zamora J Silverstone J G C Veinot andA Meldrum Refractometric sensing with fluorescent-core micro-cavities Opt Express 19 21540 (2011)

72 G Huang Q Bolanos V A Bolanos F Ding S KiravittayaY Mei and O G Schmidt Rolled-up optical microcavities withsubwavelength wall thicknesses for enhanced liquid sensing appli-cations ACS Nano 4 3123 (2010)

73 P Measor L Seballos D Yin J Z Zhang E J Lunt A RHawkins and H Schmidt On-chip surface-enhanced Raman scat-tering detection using integrated liquidndashcore waveguides ApplPhys Lett 90 211107 (2007)

74 J D Suter I M White H Zhu and X Fan Thermal characteriza-tion of liquid core optical ring resonator sensors Appl Opt 46 389(2007)

75 N Lin L Jiang S Wang H Xiao Y Lu and H-L Tsai Designand optimization of liquid core optical ring resonator for refractiveindex sensing Appl Opt 50 3615 (2011)

76 L Ding P Senellart A Lemaitre S Ducci G Leo and I FaveroGaAs micro-nanodisks probed by a looped fiber taper for optome-chanics applications Nanophotonics III edited by D L AndrewsJ-M Nunzi and A Ostendorf Proc of SPIE (2010) Vol 7712pp 771211

77 C Baker C Belacel A Andronico P Senellart A LemaitreE Galopin S Ducci G Leo and I Favero Critical optical cou-pling between a GaAs disk and a nanowaveguide suspended on thechip Appl Phys Lett 99 151117 (2011)

78 S M Spillane T J Kippenberg O J Painter and K J VahalaIdeality in a fiber-taper-coupled microresonator system for applica-tion to cavity quantum electrodynamics Phys Rev Lett 91 043902(2003)

79 A Morand Y Zhang B Martin K P Huy D Amans andP Benech Ultra-compact microdisk resonator filters on SOI sub-strate Opt Express 25 12814 (2006)

80 E F Franchimon K R Hiremath R Stoffer and M HammerJournal of the Optical Society of America B 30 1048 (2013)

81 J A Lock Morphology-dependent resonances of an infinitely longcircular cylinder illuminated by a diagonally incident plane waveor a focused Gaussian beam Journal of the Optical Society ofAmerica A 14 653 (1997)

82 A W Poon R K Chang and J A Lock Spiral morphology-dependent resonances in an optical fiber Effects of fiber tilt andfocused Gaussian beam illumination Opt Lett 23 1105 (1998)

83 V Zamora A Diacuteez M V Andreacutes and B Gimeno Refractometricsensor based on whispering-gallery modes of thin capillaries OptExpress 15 12011 (2007)

84 V Zamora A Diacuteez M V Andreacutes and B Gimeno Interrogation ofwhispering-gallery modes resonances in cylindrical microcavitiesby backreflection detection Opt Lett 34 1039 (2009)

85 A L Stancik and E B Brauns A simple asymmetric lineshape forfitting infrared absorption spectra Vibr Spectr 47 66 (2008)

86 V Zamora Z Zhang and A Meldrum Refractometric sensing ofheavy oils in fluorescent core microcapillaries Oil and Gas Scienceand TechnologymdashRev IFP Energies Nouvelles 2013113 (2013)

87 N Aissaoui L Bergaoui J Landoulsi J-F Lambert andS Boujday Silane Layers on Silicon Surfaces Mechanism of inter-action stability and influence on protein adsorption Langmuir28 656 (2012)

88 D Knopp D Tang and R Niessner Review bioanalytical appli-cations of biomolecule-functionalized nanometer-sized doped silicaparticles Anal Chim Acta 647 14 (2009)

89 J J Cras C A Rowe-Taitt D A Nivens and F S Ligler Com-parison of chemical cleaning methods of glass in preparation forsilanization Biosens Bioelectron 14 683 (1999)





90 A V Krasnoslobodtsev and S N Smirnov Effect of water onsilanization of silica by trimethoxysilanes Langmuir 18 3181(2002)

91 J A Howarter and J P Youngblood Optimization of silicasilanization by 3-Aminopropyltriethoxysilane Langmuir 22 11142(2006)

92 I M White S I Shapova H Zhu J D Suter S Lacey P ZhangH Oveys L Brewington J Gohring and X Fan Applicationsof the liquid core optical ring resonator platform Proc of SPIE6757 675707 (2007)

93 J D Suter I M White H Zhu H Shi C W Caldwell andX Fan Label-free quantitative DNA detection using the liquid coreoptical ring resonator Biosens Bioelectron 23 1003 (2008)

94 A Carreacute V Lacarriegravere and W Birch Molecular interactionsbetween DNA and an aminated glass substrate J Colloid InterfaceSci 260 49 (2003)

95 J D Sutera D J Howard H Shi C W Caldwell and X FanLabel-free DNA methylation analysis using opto-fluidic ring res-onators Biosens Bioelectron 26 1016 (2010)

96 E P Diamandis and T K Christopoulos The biotin-(strept)avidinsystem Principles and applications in biotechnology Clin Chem37 625 (1991)

97 See for example Avidin-Biotin Interactions Methods in MolecularBiology edited by R J McMahon Springer Humana Press (2008)

98 J Wong A Chilkoti and V T Moy Direct force measurements ofthe streptavidinndashbiotin interaction Biomol Eng 16 45 (1999)

99 Y L Jeyachandran J A Mielczarski E Mielczarski and B RaiEfficiency of blocking of non-specific interaction of different pro-teins by BSA adsorbed on hydrophobic and hydrophilic surfacesJ Colloid Interface Sci 341 136 (2010)

100 A S Anderson A M Dattelbaum G A Montantildeo D N PriceJ G Schmidt J S Martinez W K Grace K M Grace andB I Swanson Functional PEG-modified thin films for biologicaldetection Langmuir 2008 2240 (2008)

101 N A Lapin and Y J Chabal Infrared characterization of biotiny-lated silicon oxide surfaces surface stability and specific attach-ment of streptavidin J Phys Chem B 113 8776 (2009)

102 S McFarlane Biosensing using fluorescent-core microcapillariesUniversity of Alberta Masters Thesis (2012)

103 P S Stayton S Freitag L A Klumb A Chilkoti V ChuJ E Penzotti R To D Hyre I Le Trong T P Lybrand andR E Stenkamp Streptavidin-biotin binding energetics BiomolEng 16 39 (1999)

104 F Vollmer D Braun A Libchaber M Khoshsima I Teraoka andS Arnold Protein detection by optical shift of a resonant micro-cavity Appl Phys Lett 21 4507 (2002)

105 S McFarlane C P K Manchee J W Silverstone J G C Veinotand A Meldrum Feasibility of a fluorescent-core microcapillaryfor biosensing applications Sensor Lett 11 1513 (2013)

106 H Li and X Fan Characterization of sensing capability of optoflu-idic ring resonator biosensors Appl Phys Lett 97 011105 (2010)

107 H Zhu I M White J D Suter M Zourob and X Fan Opto-fluidic micro-ring resonator for sensitive label-free viral detectionAnalyst 133 356 (2008)

108 H Zhu I M White J D Suter and X Fan Phage-based label-freebiomolecule detection in an opto-fluidic ring resonator BiosensBioelectron 24 461 (2008)

109 A Lee G-Y Kim and J-H Moon Detection of6-benzylaminopurine plant growth regulator in bean sprouts usingOFRR biosensor and QuEChERS method Analytical Methods5 961 (2013)

110 Gilmo Yang a b Ian M White a and Xudong Fan An opto-fluidicring resonator biosensor for the detection of organophosphorus pes-ticides Sens Actuators B 133 105 (2008)

111 J T Gohring P S Dale and X Fan Detection of HER2 breastcancer biomarker using the opto-fluidic ring resonator biosensorSens Actuators B 146 226 (2010)

112 See httpgenalytecom





2 HISTORICAL BACKGROUNDThe term ldquowhispering galleryrdquo seems to have originated todescribe the peculiar acoustic effects that were reported indomed buildings such as St Paulrsquos cathedral in London30

and the Gol Gumbaz Mausoleum in northern India31

In these circular domed structures a whisper at one sideis clearly audible at the other side of the dome but itcannot be heard near the center Lord Rayleighrsquos 1910paper titled ldquoThe problem of the whispering galleryrdquo30

specifically focused on what he called an acoustic ldquocling-ing effectrdquo Essentially the ldquosonorous vibrationsrdquo producednear one wall of the cathedral appeared spatially limitedto a tiny fraction of the radius very near the cathedralwall at the other side (as measured by the flicker of acandle flame) Lord Rayleighrsquos paper reported that theBessel function Jlz which was the known solution fora circular resonance with fixed boundaries has interestingproperties if l is large In such cases he showed that theresonant vibrations must indeed ldquoclingrdquo very near to theouter radius of the resonator and are nearly zero insideexactly as observedGustav Mie had just a few years earlier developed

the theory for the scattering of light by small aerosoldroplets32 These structures were known to have electro-magnetic resonances that were similar in many respects toLord Rayleighrsquos whispering gallery modes and the nameeventually referred to both the acoustic and optical reso-nances Part of Miersquos work was based on earlier light scat-tering studies by Thomson33 Alfred Clebsch34 LudwigLorenz3536 and Peter Debye37 A complete history ofMiersquos electromagnetic scattering theory was assembled by

A Meldrum obtained his PhD in 1997 from the University of New Mexico where hestudied radiation-solid interactions in materials for nuclear waste disposal He went on toa postdoctoral fellowship at Oak Ridge National Laboratory working on the microstruc-ture and properties of semiconductor and magnetic nanocrystals Dr Meldrum joined theUniversity of Alberta as Assistant Professor in physics in September 1999 Currently hefocuses on optical cavities optical sensors fluorescence imaging and spectroscopy and thebasic physics of cavity-quantum-dot interactions He additionally works on new ways tobring research into the classrooms and to provide undergrads a greater exposure to physicsresearch Now a full professor and Vargo Teaching chair he has more than 130 refereedpublications in physics materials science optics and earth science journals

F Marsiglio graduated with a BASc from the Engineering Science program at the Uni-versity of Toronto in 1983 then went on to do graduate work in physics at McMasterUniversity where he received both a Masters and PhD (1988) Subsequently he held anNSERC postdoctoral fellowship at the University of California San Diego In 1990 hejoined AECL Research at Chalk River Laboratories During this time his work has focusedon the theory of superconductivity particularly as applied to the so-called high temperaturesuperconductors In 1997 he became an associate professor in the Department of Physicsat the University of Alberta and a full professor in 2001 Coincidentally he became theDirector of the Theoretical Physics Institute and kept this position until 2008 In 2013 hebecame Associate Chair of Research in the department He was a member of the QuantumMaterials Program in the Canadian Institute for Advanced Research His research coverssuperconductivity polarons non-equilibrium magnetism and resonances in optical cavities

Horvath38 and by Wriedt39 and these are excellent sourcesfor a more complete history of the WGMs Essentiallywhile the basic properties of optical WGMs were under-stood in the early part of the 20th century the mathematicsdescribing the Bessel functions needed in the Mie the-ory were for the most part too complicated to be fullytreated until the advent of computers Much more recentefforts gave complete modern descriptions of the sphericalWGMs4041

Cylindrical structures have a set of resonances that aresimilar to the spherical ones the main difference beingthat there is no polar component to the field which can beassumed to propagate in a 2-dimensional cylindrical crosssection similar to Rayleighrsquos acoustic modes (the effect ofwaves with some component of their wavevector along thecapillary axis will also be discussed however) The equa-tions used to model the resonances are mainly the same asfor the spherical case with the main differences being theuse of cylindrical rather than spherical Bessel functionsand the lack of a ldquopolarrdquo direction (ie one need not con-sider the Legendre harmonics) in the field solutionsThus only a few parameters are needed to completely

describe the capillary resonances the number of inten-sity maxima in the radial direction (n) the number ofwavelengths that ldquofitrdquo around the cylinder for a given res-onance (l) and the quality (Q) factor that describes howwell the mode is able to trap radiation Three such reso-nances are illustrated in Figure 1 showing different val-ues of n The polarization can be either transverse electric(TE) or transverse magnetic (TM) Unfortunately thereis no consensus as to the definition of the polarization
















2Ezr+

z

(Ezr

1

r

z

)+k20m

2Ezr= 0


2Ezr =2Ezr

z2+ 1

r22Ezr

2+ 1

r

r

times(rEzr

r

)(3)






1r2

2Ezr

2= 1

r22Ezre

il

2=minus l2

r2Ezr (4)


1r

r

r middot Ezr

r= 2Ezr

r2+ 1

r

Ezr

r(5)


eplusmnil

[r2d2Ezr

dr2+ r

dEzr


]= 0



R2 d2EzR

dR2+R

dEzR







mr=

⎧⎪⎪⎨⎪⎪⎩

m1 r le b

m2 b lt r le a

m3 r gt a

⎫⎪⎪⎬⎪⎪⎭

(8)



H1l mkr =

radic2








2l R+ClH

1l R where Bl


SR=

⎧⎪⎪⎨⎪⎪⎩

AlJlm1k0r r le b

BlH2l m2k0r+H


DlH1l m3k0r r gt a

⎫⎪⎪⎬⎪⎪⎭(10)


ErR = 0

ER = 0 (11)

EzR = SR



=BlH2l m2k0b+H

1l m2k0b (12a)


1l m2k0a

=DlH1l m3k0a (12b)


boundary 1


2l

primem2k0b

+H1l

primem2k0b (13a)

boundary 2 m2BlH2l

primem2k0a+H1l

primem2k0a

= m3DlH1l

primem3k0a (13b)






J primel nkx=

dJlmkxxrarrab

mk middotdx (14)


m3H1l

primem3k0a

m2H1l m3k0a

= BlH2l

primem2k0a+H1l

primem2k0a

BlH2l m2k0a+H

1l m2k0a

(15)


Bl=m2Jlm1k0bH

1l


1l m2k0b

minusm2Jlm1k0bH2l


2l m2k0b





Al =BlH

2l m2k0b+H

1l m2k0b

Jlm1k0b(17a)

Dl =BlH

2l m2k0a+H

1l m2k0a

H1l m3k0a

(17b)



Ez =radicReEz

2+ ImEz2 (18)




2Hzrminus1

r

Hzr

r+k20m

2Hzr= 0 (19)








T R =

⎧⎪⎪⎨⎪⎪⎩

alJlm1k0r r le b

blH2l m2k0r+H


dlH1l m3k0r r gt a

⎫⎪⎪⎬⎪⎪⎭(20)



il (21)


ErR=minus 1k0m

2

l

rT mk0r

ER=minus i

k0m2

dT mk0r

dr(22)

EzR = 0



boundary 1


2l

primem2k0b

+H1l

primem2k0b (23a)

boundary 2 1m2blH2l

primem2k0a+H1l

primem2k0a

= 1m3dlH1l

primem3k0a (23b)




m2H1l

primem3k0a

m3H1l m3k0a

= blH2l

primem2k0a+H1l

primem2k0a

blH2l m2k0a+H

1l m2k0a

(24)

and

bl=m1Jlm1k0bH

1l


1l m2k0b

minusm1Jlm1k0bH2l


2l m2k0b


(15) and (16)


























0 asymp2a

lf1m1+ f2m2+ f3m3 (26)


S = d

dm1

asymp 2af1l

asymp 0f1m2

(27)


fundamental way

S = d

dm1

= minus2dkk20

dm1

=minus 0

dm1

middot dkk0

(28)


dk

k0=minus1

2

intVpr

minusrarrElowast




3r(29)





STE = 0

m1

middot I1I1+ I2+ I3

(30a)

STM = 0

2k21

m31k

20 + klowast02

middot(bal2

(dJ lowast

l m1k0r

dr

∥∥∥∥b


times dJlm1k0r

dr

∥∥∥∥b

)+ k20+ klowast0

2

k201I1

)

middot([

I1m2

1

+ I2m2

2

+ I3m2

3

])minus1

(30b)



I1 =int m1k0 1b

0xAlJlx2 dx (31a)

I2 =int m2k01a

m2k0 1bxBlH

2l x+H

1l x2 dx (31b)

I3 =int

m3k0axDlH

1l x2 dx (31c)








d

= pex

2radicm2

2minusm21

020

m2

m21

S (32)




DL= minS (33)





SNR051meters (34)























d

dT= 0

(1

meff

dmeff

dT+ 1

a

da

dT

)(36)


dT= 0

(I1m

211+ I2m

222+ I3m

233

I1m1+ I2m2+ I3m3

+

)

(37)

dmin TE

dT= 0

(I11+ I22+ I33

m1I1+m2I2+m3I3+

)(38)



dTmiddot dT

STETM(39)












= 0

Q0

(1+ e

0

)(40)



T =(1minuse0

1+e0

)2

(41)








asymp a

l

(2meffminus

2

meff

)(42)







Pf = 2Af

[1+4

(f minus f0f

)2](43)


f = 1+B0




































Chem 15 225 (1996)











































































































































2Ezr+

z

(Ezr

1

r

z

)+k20m

2Ezr= 0


2Ezr =2Ezr

z2+ 1

r22Ezr

2+ 1

r

r

times(rEzr

r

)(3)






1r2

2Ezr

2= 1

r22Ezre

il

2=minus l2

r2Ezr (4)


1r

r

r middot Ezr

r= 2Ezr

r2+ 1

r

Ezr

r(5)


eplusmnil

[r2d2Ezr

dr2+ r

dEzr


]= 0



R2 d2EzR

dR2+R

dEzR







mr=

⎧⎪⎪⎨⎪⎪⎩

m1 r le b

m2 b lt r le a

m3 r gt a

⎫⎪⎪⎬⎪⎪⎭

(8)



H1l mkr =

radic2








2l R+ClH

1l R where Bl


SR=

⎧⎪⎪⎨⎪⎪⎩

AlJlm1k0r r le b

BlH2l m2k0r+H


DlH1l m3k0r r gt a

⎫⎪⎪⎬⎪⎪⎭(10)


ErR = 0

ER = 0 (11)

EzR = SR



=BlH2l m2k0b+H

1l m2k0b (12a)


1l m2k0a

=DlH1l m3k0a (12b)


boundary 1


2l

primem2k0b

+H1l

primem2k0b (13a)

boundary 2 m2BlH2l

primem2k0a+H1l

primem2k0a

= m3DlH1l

primem3k0a (13b)






J primel nkx=

dJlmkxxrarrab

mk middotdx (14)


m3H1l

primem3k0a

m2H1l m3k0a

= BlH2l

primem2k0a+H1l

primem2k0a

BlH2l m2k0a+H

1l m2k0a

(15)


Bl=m2Jlm1k0bH

1l


1l m2k0b

minusm2Jlm1k0bH2l


2l m2k0b





Al =BlH

2l m2k0b+H

1l m2k0b

Jlm1k0b(17a)

Dl =BlH

2l m2k0a+H

1l m2k0a

H1l m3k0a

(17b)



Ez =radicReEz

2+ ImEz2 (18)




2Hzrminus1

r

Hzr

r+k20m

2Hzr= 0 (19)








T R =

⎧⎪⎪⎨⎪⎪⎩

alJlm1k0r r le b

blH2l m2k0r+H


dlH1l m3k0r r gt a

⎫⎪⎪⎬⎪⎪⎭(20)



il (21)


ErR=minus 1k0m

2

l

rT mk0r

ER=minus i

k0m2

dT mk0r

dr(22)

EzR = 0



boundary 1


2l

primem2k0b

+H1l

primem2k0b (23a)

boundary 2 1m2blH2l

primem2k0a+H1l

primem2k0a

= 1m3dlH1l

primem3k0a (23b)




m2H1l

primem3k0a

m3H1l m3k0a

= blH2l

primem2k0a+H1l

primem2k0a

blH2l m2k0a+H

1l m2k0a

(24)

and

bl=m1Jlm1k0bH

1l


1l m2k0b

minusm1Jlm1k0bH2l


2l m2k0b


(15) and (16)


























0 asymp2a

lf1m1+ f2m2+ f3m3 (26)


S = d

dm1

asymp 2af1l

asymp 0f1m2

(27)


fundamental way

S = d

dm1

= minus2dkk20

dm1

=minus 0

dm1

middot dkk0

(28)


dk

k0=minus1

2

intVpr

minusrarrElowast




3r(29)





STE = 0

m1

middot I1I1+ I2+ I3

(30a)

STM = 0

2k21

m31k

20 + klowast02

middot(bal2

(dJ lowast

l m1k0r

dr

∥∥∥∥b


times dJlm1k0r

dr

∥∥∥∥b

)+ k20+ klowast0

2

k201I1

)

middot([

I1m2

1

+ I2m2

2

+ I3m2

3

])minus1

(30b)



I1 =int m1k0 1b

0xAlJlx2 dx (31a)

I2 =int m2k01a

m2k0 1bxBlH

2l x+H

1l x2 dx (31b)

I3 =int

m3k0axDlH

1l x2 dx (31c)








d

= pex

2radicm2

2minusm21

020

m2

m21

S (32)




DL= minS (33)





SNR051meters (34)























d

dT= 0

(1

meff

dmeff

dT+ 1

a

da

dT

)(36)


dT= 0

(I1m

211+ I2m

222+ I3m

233

I1m1+ I2m2+ I3m3

+

)

(37)

dmin TE

dT= 0

(I11+ I22+ I33

m1I1+m2I2+m3I3+

)(38)



dTmiddot dT

STETM(39)












= 0

Q0

(1+ e

0

)(40)



T =(1minuse0

1+e0

)2

(41)








asymp a

l

(2meffminus

2

meff

)(42)







Pf = 2Af

[1+4

(f minus f0f

)2](43)


f = 1+B0




































Chem 15 225 (1996)

































































































































1r2

2Ezr

2= 1

r22Ezre

il

2=minus l2

r2Ezr (4)


1r

r

r middot Ezr

r= 2Ezr

r2+ 1

r

Ezr

r(5)


eplusmnil

[r2d2Ezr

dr2+ r

dEzr


]= 0



R2 d2EzR

dR2+R

dEzR







mr=

⎧⎪⎪⎨⎪⎪⎩

m1 r le b

m2 b lt r le a

m3 r gt a

⎫⎪⎪⎬⎪⎪⎭

(8)



H1l mkr =

radic2








2l R+ClH

1l R where Bl


SR=

⎧⎪⎪⎨⎪⎪⎩

AlJlm1k0r r le b

BlH2l m2k0r+H


DlH1l m3k0r r gt a

⎫⎪⎪⎬⎪⎪⎭(10)


ErR = 0

ER = 0 (11)

EzR = SR



=BlH2l m2k0b+H

1l m2k0b (12a)


1l m2k0a

=DlH1l m3k0a (12b)


boundary 1


2l

primem2k0b

+H1l

primem2k0b (13a)

boundary 2 m2BlH2l

primem2k0a+H1l

primem2k0a

= m3DlH1l

primem3k0a (13b)






J primel nkx=

dJlmkxxrarrab

mk middotdx (14)


m3H1l

primem3k0a

m2H1l m3k0a

= BlH2l

primem2k0a+H1l

primem2k0a

BlH2l m2k0a+H

1l m2k0a

(15)


Bl=m2Jlm1k0bH

1l


1l m2k0b

minusm2Jlm1k0bH2l


2l m2k0b





Al =BlH

2l m2k0b+H

1l m2k0b

Jlm1k0b(17a)

Dl =BlH

2l m2k0a+H

1l m2k0a

H1l m3k0a

(17b)



Ez =radicReEz

2+ ImEz2 (18)




2Hzrminus1

r

Hzr

r+k20m

2Hzr= 0 (19)








T R =

⎧⎪⎪⎨⎪⎪⎩

alJlm1k0r r le b

blH2l m2k0r+H


dlH1l m3k0r r gt a

⎫⎪⎪⎬⎪⎪⎭(20)



il (21)


ErR=minus 1k0m

2

l

rT mk0r

ER=minus i

k0m2

dT mk0r

dr(22)

EzR = 0



boundary 1


2l

primem2k0b

+H1l

primem2k0b (23a)

boundary 2 1m2blH2l

primem2k0a+H1l

primem2k0a

= 1m3dlH1l

primem3k0a (23b)




m2H1l

primem3k0a

m3H1l m3k0a

= blH2l

primem2k0a+H1l

primem2k0a

blH2l m2k0a+H

1l m2k0a

(24)

and

bl=m1Jlm1k0bH

1l


1l m2k0b

minusm1Jlm1k0bH2l


2l m2k0b


(15) and (16)


























0 asymp2a

lf1m1+ f2m2+ f3m3 (26)


S = d

dm1

asymp 2af1l

asymp 0f1m2

(27)


fundamental way

S = d

dm1

= minus2dkk20

dm1

=minus 0

dm1

middot dkk0

(28)


dk

k0=minus1

2

intVpr

minusrarrElowast




3r(29)





STE = 0

m1

middot I1I1+ I2+ I3

(30a)

STM = 0

2k21

m31k

20 + klowast02

middot(bal2

(dJ lowast

l m1k0r

dr

∥∥∥∥b


times dJlm1k0r

dr

∥∥∥∥b

)+ k20+ klowast0

2

k201I1

)

middot([

I1m2

1

+ I2m2

2

+ I3m2

3

])minus1

(30b)



I1 =int m1k0 1b

0xAlJlx2 dx (31a)

I2 =int m2k01a

m2k0 1bxBlH

2l x+H

1l x2 dx (31b)

I3 =int

m3k0axDlH

1l x2 dx (31c)








d

= pex

2radicm2

2minusm21

020

m2

m21

S (32)




DL= minS (33)





SNR051meters (34)























d

dT= 0

(1

meff

dmeff

dT+ 1

a

da

dT

)(36)


dT= 0

(I1m

211+ I2m

222+ I3m

233

I1m1+ I2m2+ I3m3

+

)

(37)

dmin TE

dT= 0

(I11+ I22+ I33

m1I1+m2I2+m3I3+

)(38)



dTmiddot dT

STETM(39)












= 0

Q0

(1+ e

0

)(40)



T =(1minuse0

1+e0

)2

(41)








asymp a

l

(2meffminus

2

meff

)(42)







Pf = 2Af

[1+4

(f minus f0f

)2](43)


f = 1+B0




































Chem 15 225 (1996)

































































































































J primel nkx=

dJlmkxxrarrab

mk middotdx (14)


m3H1l

primem3k0a

m2H1l m3k0a

= BlH2l

primem2k0a+H1l

primem2k0a

BlH2l m2k0a+H

1l m2k0a

(15)


Bl=m2Jlm1k0bH

1l


1l m2k0b

minusm2Jlm1k0bH2l


2l m2k0b





Al =BlH

2l m2k0b+H

1l m2k0b

Jlm1k0b(17a)

Dl =BlH

2l m2k0a+H

1l m2k0a

H1l m3k0a

(17b)



Ez =radicReEz

2+ ImEz2 (18)




2Hzrminus1

r

Hzr

r+k20m

2Hzr= 0 (19)








T R =

⎧⎪⎪⎨⎪⎪⎩

alJlm1k0r r le b

blH2l m2k0r+H


dlH1l m3k0r r gt a

⎫⎪⎪⎬⎪⎪⎭(20)



il (21)


ErR=minus 1k0m

2

l

rT mk0r

ER=minus i

k0m2

dT mk0r

dr(22)

EzR = 0



boundary 1


2l

primem2k0b

+H1l

primem2k0b (23a)

boundary 2 1m2blH2l

primem2k0a+H1l

primem2k0a

= 1m3dlH1l

primem3k0a (23b)




m2H1l

primem3k0a

m3H1l m3k0a

= blH2l

primem2k0a+H1l

primem2k0a

blH2l m2k0a+H

1l m2k0a

(24)

and

bl=m1Jlm1k0bH

1l


1l m2k0b

minusm1Jlm1k0bH2l


2l m2k0b


(15) and (16)


























0 asymp2a

lf1m1+ f2m2+ f3m3 (26)


S = d

dm1

asymp 2af1l

asymp 0f1m2

(27)


fundamental way

S = d

dm1

= minus2dkk20

dm1

=minus 0

dm1

middot dkk0

(28)


dk

k0=minus1

2

intVpr

minusrarrElowast




3r(29)





STE = 0

m1

middot I1I1+ I2+ I3

(30a)

STM = 0

2k21

m31k

20 + klowast02

middot(bal2

(dJ lowast

l m1k0r

dr

∥∥∥∥b


times dJlm1k0r

dr

∥∥∥∥b

)+ k20+ klowast0

2

k201I1

)

middot([

I1m2

1

+ I2m2

2

+ I3m2

3

])minus1

(30b)



I1 =int m1k0 1b

0xAlJlx2 dx (31a)

I2 =int m2k01a

m2k0 1bxBlH

2l x+H

1l x2 dx (31b)

I3 =int

m3k0axDlH

1l x2 dx (31c)








d

= pex

2radicm2

2minusm21

020

m2

m21

S (32)




DL= minS (33)





SNR051meters (34)























d

dT= 0

(1

meff

dmeff

dT+ 1

a

da

dT

)(36)


dT= 0

(I1m

211+ I2m

222+ I3m

233

I1m1+ I2m2+ I3m3

+

)

(37)

dmin TE

dT= 0

(I11+ I22+ I33

m1I1+m2I2+m3I3+

)(38)



dTmiddot dT

STETM(39)












= 0

Q0

(1+ e

0

)(40)



T =(1minuse0

1+e0

)2

(41)








asymp a

l

(2meffminus

2

meff

)(42)







Pf = 2Af

[1+4

(f minus f0f

)2](43)


f = 1+B0




































Chem 15 225 (1996)

































































































































T R =

⎧⎪⎪⎨⎪⎪⎩

alJlm1k0r r le b

blH2l m2k0r+H


dlH1l m3k0r r gt a

⎫⎪⎪⎬⎪⎪⎭(20)



il (21)


ErR=minus 1k0m

2

l

rT mk0r

ER=minus i

k0m2

dT mk0r

dr(22)

EzR = 0



boundary 1


2l

primem2k0b

+H1l

primem2k0b (23a)

boundary 2 1m2blH2l

primem2k0a+H1l

primem2k0a

= 1m3dlH1l

primem3k0a (23b)




m2H1l

primem3k0a

m3H1l m3k0a

= blH2l

primem2k0a+H1l

primem2k0a

blH2l m2k0a+H

1l m2k0a

(24)

and

bl=m1Jlm1k0bH

1l


1l m2k0b

minusm1Jlm1k0bH2l


2l m2k0b


(15) and (16)


























0 asymp2a

lf1m1+ f2m2+ f3m3 (26)


S = d

dm1

asymp 2af1l

asymp 0f1m2

(27)


fundamental way

S = d

dm1

= minus2dkk20

dm1

=minus 0

dm1

middot dkk0

(28)


dk

k0=minus1

2

intVpr

minusrarrElowast




3r(29)





STE = 0

m1

middot I1I1+ I2+ I3

(30a)

STM = 0

2k21

m31k

20 + klowast02

middot(bal2

(dJ lowast

l m1k0r

dr

∥∥∥∥b


times dJlm1k0r

dr

∥∥∥∥b

)+ k20+ klowast0

2

k201I1

)

middot([

I1m2

1

+ I2m2

2

+ I3m2

3

])minus1

(30b)



I1 =int m1k0 1b

0xAlJlx2 dx (31a)

I2 =int m2k01a

m2k0 1bxBlH

2l x+H

1l x2 dx (31b)

I3 =int

m3k0axDlH

1l x2 dx (31c)








d

= pex

2radicm2

2minusm21

020

m2

m21

S (32)




DL= minS (33)





SNR051meters (34)























d

dT= 0

(1

meff

dmeff

dT+ 1

a

da

dT

)(36)


dT= 0

(I1m

211+ I2m

222+ I3m

233

I1m1+ I2m2+ I3m3

+

)

(37)

dmin TE

dT= 0

(I11+ I22+ I33

m1I1+m2I2+m3I3+

)(38)



dTmiddot dT

STETM(39)












= 0

Q0

(1+ e

0

)(40)



T =(1minuse0

1+e0

)2

(41)








asymp a

l

(2meffminus

2

meff

)(42)







Pf = 2Af

[1+4

(f minus f0f

)2](43)


f = 1+B0




































Chem 15 225 (1996)















































































































































0 asymp2a

lf1m1+ f2m2+ f3m3 (26)


S = d

dm1

asymp 2af1l

asymp 0f1m2

(27)


fundamental way

S = d

dm1

= minus2dkk20

dm1

=minus 0

dm1

middot dkk0

(28)


dk

k0=minus1

2

intVpr

minusrarrElowast




3r(29)





STE = 0

m1

middot I1I1+ I2+ I3

(30a)

STM = 0

2k21

m31k

20 + klowast02

middot(bal2

(dJ lowast

l m1k0r

dr

∥∥∥∥b


times dJlm1k0r

dr

∥∥∥∥b

)+ k20+ klowast0

2

k201I1

)

middot([

I1m2

1

+ I2m2

2

+ I3m2

3

])minus1

(30b)



I1 =int m1k0 1b

0xAlJlx2 dx (31a)

I2 =int m2k01a

m2k0 1bxBlH

2l x+H

1l x2 dx (31b)

I3 =int

m3k0axDlH

1l x2 dx (31c)








d

= pex

2radicm2

2minusm21

020

m2

m21

S (32)




DL= minS (33)





SNR051meters (34)























d

dT= 0

(1

meff

dmeff

dT+ 1

a

da

dT

)(36)


dT= 0

(I1m

211+ I2m

222+ I3m

233

I1m1+ I2m2+ I3m3

+

)

(37)

dmin TE

dT= 0

(I11+ I22+ I33

m1I1+m2I2+m3I3+

)(38)



dTmiddot dT

STETM(39)












= 0

Q0

(1+ e

0

)(40)



T =(1minuse0

1+e0

)2

(41)








asymp a

l

(2meffminus

2

meff

)(42)







Pf = 2Af

[1+4

(f minus f0f

)2](43)


f = 1+B0




































Chem 15 225 (1996)

































































































































d

= pex

2radicm2

2minusm21

020

m2

m21

S (32)




DL= minS (33)





SNR051meters (34)























d

dT= 0

(1

meff

dmeff

dT+ 1

a

da

dT

)(36)


dT= 0

(I1m

211+ I2m

222+ I3m

233

I1m1+ I2m2+ I3m3

+

)

(37)

dmin TE

dT= 0

(I11+ I22+ I33

m1I1+m2I2+m3I3+

)(38)



dTmiddot dT

STETM(39)












= 0

Q0

(1+ e

0

)(40)



T =(1minuse0

1+e0

)2

(41)








asymp a

l

(2meffminus

2

meff

)(42)







Pf = 2Af

[1+4

(f minus f0f

)2](43)


f = 1+B0




































Chem 15 225 (1996)








































































































































d

dT= 0

(1

meff

dmeff

dT+ 1

a

da

dT

)(36)


dT= 0

(I1m

211+ I2m

222+ I3m

233

I1m1+ I2m2+ I3m3

+

)

(37)

dmin TE

dT= 0

(I11+ I22+ I33

m1I1+m2I2+m3I3+

)(38)



dTmiddot dT

STETM(39)












= 0

Q0

(1+ e

0

)(40)



T =(1minuse0

1+e0

)2

(41)








asymp a

l

(2meffminus

2

meff

)(42)







Pf = 2Af

[1+4

(f minus f0f

)2](43)


f = 1+B0




































Chem 15 225 (1996)






































































































































= 0

Q0

(1+ e

0

)(40)



T =(1minuse0

1+e0

)2

(41)








asymp a

l

(2meffminus

2

meff

)(42)







Pf = 2Af

[1+4

(f minus f0f

)2](43)


f = 1+B0




































Chem 15 225 (1996)


































































































































asymp a

l

(2meffminus

2

meff

)(42)







Pf = 2Af

[1+4

(f minus f0f

)2](43)


f = 1+B0




































Chem 15 225 (1996)



























































































































































Chem 15 225 (1996)





























































































































Capillary-Type Microfluidic Sensors Based on Optical ...

Documents

Transcript of Capillary-Type Microfluidic Sensors Based on Optical ...