Advances in micro- and nanofabrication techniqueshave made it feasible to consider optical resonatorshaving physical or feature dimensions of the order ofone optical wavelength or less. As a particular modeof microcavity resonances,1 whispering-gallerymodes (WGMs) occur at specific wavelengths whenlight rays travel in a dielectric medium of circulargeometry such as spheres, disks, rings, and cylinders.The resonance occurs when the electromagnetic (EM)field closes on itself at the curvilinear boundary afterrepeated total internal reflections. Since the pioneer-ing work2 in the excitation of WGM resonances ofmicrospheres on an optical fiber, microcavity WGMresonators have attracted increasing attention in re-search and technology development in recent years
due to their high potentials for realization of micro-lasers,3 narrow filters,4 optical switching,5 single mol-ecule detection biosensors,6 and high-resolutionspectroscopy.7
As stressed by Arnold,1,8 WGMs are morphology-dependent resonances. The resonant frequencies de-pend on the size of the resonator. In general, theresonant modes for a circular resonator are approxi-mately predicted by 2�rn � mc0�f, where m is aninteger, n is the refractive index of the cavity mate-rial, r is the radius of the cavity, and f is the resonantfrequency of mode m. The frequency shift of a givenresonant mode is estimated to be �f�f � ���r�r� �n�n�, where �r and �n represent small changes ofthe radius and refractive index of the cavity, respec-tively. If we assume a constant refractive index andconsider the linewidth of the resonance to be thesmallest measurable shift (taken as �f � 10 MHz,f � 3.75 � 108 MHz at � � 800 nm), then the smallest“measurable” size change is |�r|min � 2.6 � 10�8r.With a radius r in the range of 1–10 �m in a typicalmicrocavity, |�r|min is down to the order of 10�4 nm,which is an order of magnitude smaller than the sizeof an atom that is potentially detectable in theory.
Such a feature is being explored for use as detectors
Z. Guo ([email protected]) and H. Quan are with the Depart-ment of Mechanical and Aerospace Engineering, Rutgers, theState University of New Jersey, Piscataway, New Jersey 08854. S.Pau is with Nanofabrication Research Laboratory, Lucent Tech-nologies�Bell Labs, Murray Hill, New Jersey 07974.
and sensors for identifying molecules surroundingthe peripheral surface of WGM resonators. Whenpeptides, protein molecules, or cell membranes areattached on a resonator, for example, they interactwith the evanescent radiation field around the reso-nator. The interactions polarize the molecules or tar-get analytes and change the effective size and�orrefractive index of the resonator. All these can lead toa detectable frequency shift in the resonance modes.Thus it is possible to identify and detect single mol-ecules by observing the resonant frequency shifts inWGM-based optical sensors.8–10 These optical reso-nance techniques can also be used to enhance theexisting sensitivity of biosensor devices.11,12
To optimize the sensitivity of specific molecules de-tection, we must design a cavity configuration withhigh finesse. Experimental methods for conductingsuch a task are very time consuming and costly. An-alytical models13 have been introduced to analyzeoptical resonant phenomena associated with smallparticles, such as the perturbation model.14 Analyti-cal solutions are very useful and powerful in under-standing the physical essence of the phenomena.Although they can reveal the individual intuitive res-onance properties of a microcavity, it is hard for themto capture a completely real picture of a sensor as asystem. For example, a perturbation theory is hardlyable to account for the coupling of the evanescentfields in the nanoscale gap and the interactions of theresonator with surrounding individual molecules. Asa matter of fact, the EM field in the microcavity isvery sensitive against the gap through which photonstunnel. A complete modeling of the EM field in thewhole WGM structure is highly desired. A flexiblenumerical characterization can be developed into apractical tool for system and device design and opti-mization.
Previously, many WGM-based sensors were con-structed of a microsphere and an eroded optical fibercoupling structure. Although the Q value for amicrosphere-based resonator can be very high, such aconfiguration may have some flaws for use as an idealsensor. For instance, mass manufacturing of suchdevices can be difficult, and nonuniformity exists,especially in the control of the gap distance separat-ing the light-delivery fiber and the resonator. The gapis a critical parameter for photon tunneling and af-fects the Q value and resonant frequencies, as will beshown in Section 4.
Here we consider sensors of a planar waveguideand microdisk coupling structure. Such devices canbe manufactured on silicon-based thin films usingconventional silicon integrated circuits (ICs) process-ing with high uniformity and density. This new cavitystructure will further reduce the sensor size and en-hance miniaturization of the devices. Planar WGMsensors possess a high sensitivity, a small samplevolume, and a robust integrated property for system-on-a-chip applications.
Maxwell’s EM theory can be adopted to rigorouslydescribe the radiation–matter interactions in planarWGM microcavities. In this paper, Maxwell’s equa-
tions are solved using the finite-element method. Thecharacterization of the WGM-based devices is focusedon the optical resonant phenomena with respect to abroad range of cavity configuration parameters in-cluding the microcavity size, the gap width, and thewaveguide width. The effects of these parameters willbe scrutinized. The resonant frequencies are chosenin the near infrared range, which is ideal for appli-cations to biomaterials and biomolecules.
2. Electromagnetic Theory
The sketch of a planar waveguide and microdisk de-vice is shown in Fig. 1. The EM field in the WGMdevice is governed by time-dependent Maxwell’sequations. By introducing time-harmonic waves, theMaxwell’s equations can be reduced to two Helmholtzequations as follows:
1�
2E� � 2�cE� � 0, (1a)
1�
2H� � 2�cH� � 0, (1b)
where E� and H� are the electric and magnetic fieldvectors, respectively, and � 2�c��. We have intro-duced the complex permittivity �c � �cr�0 � �� i����, where �cr is the complex relative permittiv-ity and �0 is the permittivity in vacuum. ε is thepermittivity of the medium, � is the permeability,and � is the electrical conductivity. c is the speed oflight in the medium and � is the light wavelength.The relationship between �cr and m is expressed by15
�cr � m2 � n2 � k2 � i2nk. Here the complex index ofrefraction, m � n � ik, is conveniently introduced; nis the real part of the refractive index and representsa spatial phase change of the electromagnetic wave; kis the absorption index and stands for a spatial damp-ing of the electromagnetic wave.
WGM resonance inside the planar microdisk is typ-ically an equatorial brilliant ring, and this ring islocated on the same plane as the waveguide. So it isfeasible to use a two-dimensional (2D) theoreticalmodel. In the present calculations we apply in-planeTE waves, where the electric field vector has only a zcomponent and it propagates in the x–y plane. Thusthe fields can be written as
Fig. 1. Sketch of a waveguide–microdisk coupling WGM reso-nator.
To get a full description of the electromagneticproblem, we also need to specify the boundary condi-tions. At the interface and physical boundaries, weused the natural continuity condition for the tangen-tial component of the magnetic field:
n� � H� � 0. (3)
For the outside boundaries, the low-reflecting bound-ary condition is adopted. The low reflection meansthat only a small part of the wave is reflected, andthat the wave propagates through the boundary al-most as if it were not present. This condition can beformalized as
e�z · n� � ��H� � ��Ez � 0. (4)
The laser excitation source E0z, which propagatesinwards through the entry boundary of thewaveguide, can be treated as a low-reflecting bound-ary condition and can be expressed by
E0z �1
2��(e�z · n� � ��H� � ��Ez). (5)
The WGM resonances have high-quality factorsdue to minimal reflection losses. The quality factor Qis defined as a ratio of 2� stored energy to energy lostper cycle. From the energy conservation and reso-nance properties, we can deduce a simple approxi-mate expression16:
Q � 0�� � 2�0 , (6)
where 0 is the resonant frequency, � is the reso-nance linewidth, and � is the photon lifetime.
To find the radiation energy conservation, Poynt-ing’s theorem17 is employed:
�S(E� � H� ) · n� dS � ��V
�E� ·�D�
�t � H� ·�B�
�t �dV
��V
J� · E� dV, (7)
where V is the computation element volume and S isthe closed boundary surface area of V. The term onthe left-hand side of Eq. (7) represents the radiativelosses. The quantity S� � E� � H� is called as thePoynting vector. The first integral on the right-handside represents the rate of change in total energy. Thesecond integral on the right-hand side represents theresistive losses that result from heat dissipation in
metallic materials. For dielectric materials with neg-ligible absorption index, we assume zero electric cur-rent density, i.e., J� � 0. Thus the change in totalenergy of the EM field is totally converted to radiativeenergy.
3. Simulation Model
More than 30 years ago, Silvester18 developed high-order Lagrange elements and first applied the finite-element method (FEM) for solving EM field problems.Recently, Quan and Guo10 successfully applied theFEM to simulate the EM and radiation energy fieldsin the WGM resonators of a microsphere and an op-tical fiber coupling structure. Although the finite-difference time-domain (FDTD) method19 has beencommonly adopted in computational electrodynam-ics, the FEM has advantages in terms of the treat-ment of irregular configurations. This is very usefulfor simulation-based optimal design purposes.
In the present computations, the FEM is employedfor solving the Helmholtz equations. Detailed de-scription of the solution scheme has been given in arecent paper,10 and thus is not repeated here. FEMLAB
is used for finite-element analysis and pre- and post-processing.
Silicon nitride �Si3N4� is selected as the material forthe waveguide and microdisk because this substancehas excellent physical and thermal stability, low cost,and extremely low optical absorption around the op-erating wavelengths.20,21 The thickness of the Si3N4thin film is 1.3 �m. A 3 �m thick layer of SiO2 isemployed as the low cladding of the device. Thesethin films can be deposited on the surface of siliconwafer by using low-pressure chemical-vapor deposi-tion or plasma-enhanced chemical-vapor deposition.The large refractive index of Si3N4 ensures high con-trast of refractive indices between the WGM resona-tor and its surrounding medium (gas phase oraqueous solutions) and can result in high-quality res-onance modes. Figure 2 shows scanning electron mi-croscopy (SEM) photos of a fabricated such deviceusing 248 nm optical lithography and silicon IC pro-cessing.
A typical simulation domain is a 20 �m � 25 �mrectangular area with a centered microdisk (as thecavity) and a waveguide (for light delivery) below themicrodisk. The microdisk and waveguide are sepa-rated by a small gap. The length of the waveguide isextended to the edge of the simulation domain. Alaser beam from a tunable continuous-wave (CW) la-ser is coupled into the left end of the waveguide toexcite the resonance. The frequency of the incidentlaser varies between 364 THz (824 nm) and 376 THz(798 nm). When the frequency of the input light is thesame as a natural resonant frequency of the cavity,the WGM is excited. At the resonant frequency, thescattering intensity from the microdisk will increasesharply and form a peak in the intensity–frequencyspectrum.
The computational domain is meshed by 51,400triangle elements. Since fine meshes are required in
the vicinity around the periphery of the cavity, hier-archical meshing is employed to scale the cavity downto two different spatial levels (this is why a solidinternal circle is observed in Figs. 3 and 4 in Section4). The general computational resolution of the exci-tation wavelength is 0.5 nm, but special attention ispaid to the resonance frequency vicinities where0.01 nm resolution is adopted. To conduct parametricstudies, the diameter of the microdisk varies between10 and 15 �m. The width of the waveguide changes
between 2 and 3 �m. The gap distance between themicrodisk and waveguide varies between 100 and 300nm.
4. Results and Discussion
First we investigate the EM fields and radiation en-ergy distributions under WGM resonances and offresonance, respectively, and observe the differencesunder various operation conditions. Figures 3 and 4exhibit the distributions of the electric field and theradiation energy density, respectively, for three dif-ferent operating conditions. Comparisons can be per-formed between the off resonance, the first-orderresonance, and the second-order resonance. The first-and second-order resonance frequencies are found at373.78 THz (� � 802.61 nm) and 372.96 THz (804.37nm), respectively. The off-resonance frequency is se-lected at 372.67 THz (� � 805.00 nm). The diameterof the microdisk is 15 �m and the surrounding me-dium is air. The gap width, which is defined as thesmallest distance between the waveguide and micro-disk, is g � 230 nm and the width of the waveguide isw � 2 �m. The refractive index of the cavity andwaveguide material is assumed to be constant at 2.01at the operating wavelengths.21 These general para-metric values are used throughout the paper unlessotherwise specified.
In Fig. 3 we see that the EM field exists in themicrodisk even when resonant phenomenon does notoccur. Photons tunnel from the waveguide to the mi-crodisk because the gap width is less than one opticalwavelength. For the first-order resonance, we see abuildup of EM field forming inside the microdisk inthe vicinity close to the peripheral surface. Thestrength of the electric field in the ring is strongerthan that in the waveguide through where the exci-tation light is delivered. While in the case of thesecond-order resonance, there are two bright ringsinside the microdisk. The strength of the EM field inthe inner ring is stronger than that in the outer ring.Under the off-resonance condition, however, the EMfield is confined in the waveguide and its strength inthe cavity is very weak.
From Fig. 4 it is seen that the microdisk andwaveguide coupling resonator has an appealing prop-erty of high-energy storage in the cavity when WGMsoccur. The majority of the energy stores in the thinring inside the peripheral surface of the microdisk
Fig. 2. SEM photos of a fabricated WGM microcavity.
Fig. 3. Electric fields under (a) the first-order res-onance, (b) the second-order resonance, and (c) offresonance.
under first-order resonance. For second-order reso-nance, the energy is stored in two rings and the lightintensity in the inner ring is stronger. The ratio of theradiation energy storing in the microdisk to the en-ergy passing through the waveguide is found to be10.5 for the first-order resonance, whereas this ratiois only 0.008 for the off-resonance case. The increaseof energy storage in the small volume cavity leads toenormous enhancement of the micro- and nanoscaleradiation fields around the periphery of the resona-tor, which can potentially be used to sense any exter-nal perturbation.
Figure 5 shows the scattering spectra for threedifferent microdisk diameters: d � 10, 12.5, and15 �m, respectively. Three first-order resonant fre-quencies (modes) are found for each of the microdisksizes in the frequency range considered (364–376THz). We find that the microdisk size affects signifi-cantly the resonant frequencies and their intervals.The free spectral range (FSR), which represents theperiodicity of resonance peaks, increases with thedecreasing diameter of the microdisk. The FSR of theresonant modes is 3.273 � 0.007 THz for the case ofd � 15 �m, 3.940 � 0.009 THz for the case of d �12.5 �m, and 4.954 � 0.010 THz for the case of d� 10 �m. Such wide FSRs make the WGM microcavi-ties excellent candidates for detection and distin-guishing of trace gas and molecules using thespectroscopy method.
To scrutinize the effects of cavity configurations,
the resonance data retrieved from the scatteringspectra in Fig. 5 are listed in Table 1. The resonancedata include the resonant frequency and its corre-sponding wavelength, the quality factor, the fullwidth at half-maximum (FWHM) of the resonant fre-quency band, the FSR, and the finesse of the resonantmode defined by F � FSR�FWHM. To inspect theeffects of gap width and waveguide width, we obtainthe scattering spectra for three different gap widths(g � 100, 200, and 300 nm, respectively) and threedifferent waveguide widths (w � 2.0, 2.5, and3.0 �m, respectively). These are also listed in Table 1.
From Table 1 it is clear that the resonant frequen-cies are predominantly determined by the size of themicrodisk. The gap width also affects the resonantfrequencies although its effect is in a finite range of0.1 THz (of the order of 0.1 nm in terms of wave-length) for the four different gap widths considered.Such an effect can be attributed to the gap influenceon the orientation of photon tunneling. The influenc-ing range of the waveguide on the resonant frequen-cies for the waveguide widths considered is not morethan 0.01 THz, which is almost comparable to thecomputation resolution. Thus the waveguide widthmay have negligible effect on the resonant frequen-cies.
The path length of the m-mode resonant photonsorbiting the periphery of the cavity is estimated to beapproximately 2mr sin���m�. Thus the resonant fre-quencies can be analytically approximated as
f �c0
2rn sin(��m). (8)
The analytically estimated frequencies using Eq. (8)are compared with the numerically predicted fre-quencies in Fig. 6. Although the two methods giveconsistent results, it is seen that the difference of theresonant frequencies between the two methods couldbe as large as 1 THz. Nevertheless, Eq. (8) is simpleand can give a reasonable estimation of the resonantfrequencies.
Figure 7 further portrays the gap effects on thequality factor Q and finesse F of the resonances. Fourdifferent gap widths of 100, 200, 230, and 300 nm areselected for comparison. Since there are three reso-nant modes in the frequency range considered, threesets of Q values and two sets of FSR data are obtained
Fig. 4. Energy distributions under (a) the first-order resonance, (b) the second-order resonance,and (c) off resonance.
Fig. 5. Scattering spectra for different microdisk sizes of d� 10, 12.5, and 15 �m, respectively.
and displayed. With the increase of the gap widthfrom 100 to 300 nm, it is seen that both the Q and Fvalues increase by over an order of magnitude. Ex-amining Table 1, it is found that the gap width doesnot obviously affect the FSR, but does strongly influ-ence the FWHM of the resonant bands. The larger thegap width, the narrower the FWHM. In terms of thequality factor and FWHM, the effect of the microdisksize is very slight.
The effects of the waveguide width on the Q valueand the FWHM are shown in Fig. 8 for the threeresonant modes. With the decrease in the waveguidewidth, the FWHM of the resonance generally in-creases, but the Q value decreases. From Table 1 it isobvious that the waveguide width does not influencethe FSR. Comparing Figs. 7 and 8, we see that thevalue of the gap has a much larger effect on theresonance than the value of the waveguide width. Inthis paper, we have just selected several typical val-ues for the gap width. Since the gap could be designedto up to one optical wavelength or down to zero (whenthe resonator and waveguide are in close contact), amore detailed scrutiny of the gap effect in a widerspectrum of the gap width is needed and will be thefocus of our next report.
5. Conclusions
The characteristics of planar WGM microcavitieswith a waveguide–microdisk coupling structure wereinvestigated numerically. The EM fields and radia-tion energy distributions in the devices were obtainedthrough the solution of Maxwell’s equations. It wasfound that photon tunneling between the waveguideand microdisk is very weak under the off-resonancecondition and the radiation energy is well confinedinside the waveguide. When WGM resonance occurs,photon tunneling is greatly enhanced and significantradiation energy is stored in the microcavity. A verybrilliant ring with a strong EM field and a high ra-diation intensity exists inside the periphery of the
microdisk under first-order resonance, whereas thereare two bright rings inside the cavity under thesecond-order resonance, the strength of the outer ringis weaker than that of the inner ring. Thus the first-order resonances may be preferred in sensing appli-cations because the interactions of interest occur inthe evanescent field surrounding the cavity periph-ery. The WGM resonant frequencies are predomi-nantly determined by the size of the cavity. The gapseparating the waveguide and the cavity has alsoinfluence on the resonant frequencies. But thewaveguide size has negligible effect on the resonantfrequencies. The Q value is substantially influencedby the gap width. The difference in Q values between100 and 300 nm gaps is over 1 order of magnitude.The waveguide width also affects the Q value. In-creasing the waveguide width will increase the Qvalue. The FSR of the resonances reaches to severalTHz for the considered WGM microcavities. It ismainly decided by the size of the cavity and littleaffected by the gap and waveguide widths. TheFWHM of the resonances is strongly influenced bythe gap width. A wider gap results in a narrowerFWHM. The increase of the waveguide width leads tothe reduction of the FWHM. The finesse of the reso-nances is affected by the gap and waveguide widthsand the size of the cavity. Among these parameters,the gap width influences the finesse the most. Withan increase of the gap width from 100 to 300 nm, theF value increases substantially. The finesse also in-creases as the cavity size decreases or the waveguidewidth increases.
Z. Guo acknowledges the partial support of theAcademic Excellence Fund Award from Rutgers Uni-versity, the New Jersey Nanotechnology Consortium,and a National Science Foundation grant (CTS-0318001) to the project.
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