Abstract

A full three-dimensional Finite-Difference Time-Domain (FDTD)-based toolkit is developed to simulate the whispering gallery modes of a microsphere in the vicinity of a dipole source. This provides a guide for experiments that rely on efficient coupling to the modes of microspheres. The resultant spectra are compared to those of analytic models used in the field. In contrast to the analytic models, the FDTD method is able to collect flux from a variety of possible collection regions, such as a disk-shaped region. The customizability of the technique allows one to consider a variety of mode excitation scenarios, which are particularly useful for investigating novel properties of optical resonators, and are valuable in assessing the viability of a resonator for biosensing.

© 2015 Optical Society of America

1. Introduction

Whispering gallery modes (WGMs) are produced in microresonators by electromagnetic waves that travel along the material interface. Though the WGMs are bound waves, they produce an evanescent field that extends beyond the surface of the resonator. WGMs represent an important optical phenomenon for sensing due to the sensitivity of their evanescent field to nearby entities, such as biomolecules, which break the symmetry of the field [1]. Optical microresonators have been the subject of much recent investigation since their discovery as an important tool for biological sensing [2]. It has been shown that microresonator WGMs can be sensitive to the presence of virions, animal cells, bacteria [3] and macromolecules such as proteins [2,46] and DNA [7, 8], which may be used in the development of label-free detection technologies [9, 10]. WGMs have also been used for the development of high-efficiency optical frequency combs [11], investigating nonlinear optics [12] and quantum electrodynamics [13]. The focus of this study is on the development of a computational modeling tool for investigating the properties of WGMs in microspheres. In particular, the advantages and limitations of the method will be discussed.

The extremely high Q-factors (> 108) that are possible by exciting the WGMs of a resonator often utilize a prism, or a tapered fiber coupled to a resonator [1418]. An alternative approach is to excite the WGMs indirectly, by using fluorescent dyes [19], surface plasmon effects [9,20] or nanoparticle coatings such as quantum dots [21]. Here, the focus is on coupling strategies that involve the excitation of WGMs of microspheres using dipole sources. The Finite-Difference Time-Domain (FDTD) computational tool developed in this work will address this coupling scenario in particular. It should be noted, however, that the tool is easily extended to include fiber-coupled methods.

The use of dipole sources in the vicinity of a microresonator allows one to investigate the dependence of the optical modes on the size, shape and refractive index contrast. Efforts to characterize the optical properties of WGMs in resonators using modeling techniques include both excitation from a plane-wave beam [2224], and excitation from an electric dipole source [2533]. The dipole sources can also serve as an effective analogue for fluorescent dyes or embedded nanoparticles that excite WGMs in microspheres [19].

The Finite-Difference Time-Domain technique simulates the evolution of electromagnetic fields by discretizing a volume into a three-dimensional spatial lattice [34]. Maxwell’s Equations are then solved for every point on the lattice, for a finite number of time increments. The geometry of the resonator is defined for a dielectric medium, and placed in the discretized volume, with the edges padded with a field-absorbing perfectly-matched layer. FDTD is chosen specifically for its ability to incorporate effects such as material inhomogeneities, and a variety of resonator shapes, such as shells, ellipsoids or shape deformations to a microsphere.

The use of two-dimensional FDTD simulations to describe the resonance peak positions has been investigated in the case of microdisks [35]. However, the accurate prediction of the Q-factor of the resonances represents a principal challenge [36], due to the significant dependence of the Q-factor on minute characteristics of the resonator, such as surface roughness, material inhomogeneities and microscopic deformations in the shape of the resonator [20]. As a result, theoretical Q-factors evaluated through analytic models can be difficult to obtain experimentally [20, 37]. The flexibility of the FDTD method is the ability to address this issue by incorporating geometric, material and refractive index features in a way that is intuitive and easy to implement.

The FDTD method will be used to consider a variety of flux collection scenarios, source distribution, and resonator properties. In Section 4, an example scenario of a polystyrene microsphere in a surrounding medium of air or water is considered. The Q-factors obtained from the simulation are then compared to those of the Chew analytic model [30]. The angular distribution of the flux density is also investigated.

Analytic models describing WGMs in the literature generally fall into two categories: models that provide only the positions of the TE and TM modes and are unable to predict the profile of the output spectrum [38, 39], and models that consider the full behavior of the electric and magnetic fields both inside and outside the resonator, for an incident plane-wave (Mie scattering) [40] or dipole source [2533]. Although the simpler analytic models used in the literature are comparatively efficient to calculate, the profile of the transverse electric and magnetic (TE and TM) modes are obtained independently, for example, through geometrical arguments [38, 39]. This means neighboring modes do not interact with each other in the model. Both types of analytic model rely on the assumptions that the sphere is perfectly round with no surface roughness, that it consists of a medium that is homogeneous in refractive index, and that the total radiated power is collected in the limit of long collection times. The formulae for the power output are cumbersome, and must be re-derived for scenarios that consider different resonator shapes or inhomogeneities.

On the other hand, FDTD is easy to customize, as changes to the resonator geometry (spheroids, shells, etc.), refractive index scenario, and introducing material inhomogeneities can be accommodated without re-deriving the boundary conditions of the resonator. This method is able to achieve robust predictions for preselecting specific optical properties of a given resonator prior to fabrication. For example, specific modes or wavelength regions can be reliably identified, and tailored for specific tasks by altering the initial configuration of the resonator. Furthermore, the ability to scan over a wide range of parameters may lead to valuable design solutions for biosensing that would not otherwise have been found.

2. FDTD method

In this article, the three-dimensional FDTD method is simulated using the free software package, MEEP [41]. Since the simulation includes the complete set of radiation and bound modes occurring in the system, the only simulation artifacts in an FDTD calculation involve discretization and finite-volume effects, and any assumptions of ideal material properties one might make. By making the grid size finer, the simulation approaches a converged result, and this can be quantified in the form of a systematic uncertainty in the positions of the WGMs (see Section 4).

The FDTD method can lead to the possibility of accessing transient or emergent optical effects, as a Gaussian source is used, and each time step is evaluated separately. However, FDTD is computationally expensive. For example, time steps totaling a few hundred wave periods frequently require up to 100 hours of simulation time, for ∼ 30 CPUs on modern supercomputers. Table 1 summarizes typical computing resources for a range of resolutions, using the Tizard machine at eResearchSA (https://www.ersa.edu.au/tizard). Nevertheless, for each simulation run, a large amount of data can be extracted by accessing the electric, magnetic and Poynting vector fields at each grid point. This allows one to map out the angular distribution of the energy density for a typical flux plane (see Section 4).

Tables Icon

Table 1. Computing resources required for a three-dimensional FDTD simulation of a 6 μm diameter sphere excited by a dipole source with a central wavelength of 0.6 μm. The Tizard machine at eResearchSA (https://www.ersa.edu.au/tizard) is used in these simulations, which uses AMD 6238, 2.6 GHz CPUs. The number of CPUs, the memory (RAM), virtual memory (VM) and wall-time are listed for a variety of FDTD grid resolution values, Δx. The resolution in the frequency domain is quoted after being converted to wavelength, Δλ. The wall-time increases linearly with the flux collection time, which is held fixed at 0.6 ps in this table.

FDTD is also able to consider any desired position or alignment of dipole sources, including inhomogeneous distributions of dipoles placed on the surface or throughout the medium. The freedom to define a specific flux collection region at any point in space, or for any length of time, is an important feature for ongoing investigations into a range of coupling scenarios. Measuring the flux from a particular direction and aperture automatically biases the shape of the collected power spectrum. FDTD is therefore especially suitable for simulating the effect of collecting radiation through a restricted region.

The principal drawbacks in the FDTD method are the computational intensity, and systematic effects due to the discretization of space. The accuracy of each FDTD calculation is limited by the Yee cell size, which determines the grid resolution of the simulated volume. In the following simulations, the cell width is 20–30 nm. A decrease in the cell size increases the computing time requirement to the fourth power (three spatial dimensions, and time) [41]. As a result, the available hardware resources typically limit the improvement in accuracy of the FDTD simulations. This effect can, however, be incorporated into the systematic uncertainty of the calculation.

To generate an electromagnetic current to excite the WGMs, one or several electric dipole sources are placed in the vicinity of the sphere. Consider the example of a microsphere with a diameter of 6 microns, as illustrated in Fig. 1. An electric dipole is placed on the surface of the sphere and oriented tangentially to the surface, while a circular flux collection region is defined a distance Lflux, normal to the sphere, with diameter Dflux. This region aggregates the flux in the frequency domain. One may vary the orientation and position of both source and collection region to build up a map of the coupling efficiencies to different WGMs.

 

Fig. 1 A circular flux collection region, which is offset from a 6 μm diameter microsphere in the x-direction, is placed so that its normal is aligned radially outward along the same axis. In this illustration, the optical modes are excited from a tangentially-oriented electric dipole source.

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The power output spectrum represents an important quantity for assessing the Q-factors and the wavelength positions of the excited WGMs. The total output power (P) in terms of wavelength (λ) is obtained by integrating the projection of the Poynting vector (SE × H*) onto a flux region of area A:

P(λ)=Sn^dA,
for a unit vector normal to the collection region. The profile of the power spectrum, plotted as a function of λ, will show sharp peaks that correspond to the positions of TE and TM WGMs. Note that the convention used for the TE and TM mode definitions is identical to that found in [38, 39] in which the TE (TM) mode indicates that the electric (magnetic) field is tangent to the surface of the sphere. The relative heights of the peaks indicate the coupling efficiencies to different modes, which is highly dependent on the geometry of the dielectric medium, the refractive index contrast, and also the method of mode excitation, such as dipole sources of different alignments, coupling to optical fibers or fluorescent dye coatings. The power spectrum is normalized (in Figs. 2 through 4, and Fig. 7) to the power output P0 of the sources that would occur in the absence of the dielectric sphere. Note that the quantity P/P0 is proportional to the emission rate ratio, Γ/Γ0 [42].

 

Fig. 2 (a) A comparison of the power spectra of 6 μm diameter microspheres for different grid resolutions. (b) The convergence of the position of three central peaks is shown as a function of resolution. Excitation occurs from a tangentially oriented dipole source placed on the surface, with a central wavelength of λ = 0.6 μm. The flux collection time for each simulation is 0.6 ps.

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Fig. 3 (a) A comparison of the power spectra of 6 μm diameter microspheres with a tangential source (λ = 0.6 μm), for different flux collection times. (b) The behavior of the Q-factors of three central peaks is shown as a function of collection time (ps). The output power is normalized to the dipole emission rate in an infinite bulk medium of the same refractive index as the surrounding medium.

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Fig. 4 FDTD simulation of the normalized power spectrum of a polystyrene microsphere, 6 μm in diameter, with a surrounding medium of air. (a) Whispering gallery modes are excited from a tangential or (b) radial electric dipole source with a central wavelength of 0.6 μm. (c) The results for a surrounding medium of water are also shown for a tangential source and (d) a radial source. Vertical lines indicate predictions of the TEm,n (green) and TMm,n (red) modes derived from the Mie scattering analytic model, for azimuthal and radial mode numbers m and n, respectively. The width of the bands indicates the systematic uncertainty in the positions due to the finite grid size of FDTD.

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Fig. 5 A comparison of the Chew model of Eqs. (15) & (16) (dashed purple line) with the Mie scattering analytic model (vertical dashed lines) and the FDTD simulation, with total power collected from a box that surrounds the microsphere (solid blue line). A surrounding medium of water is used for a 6 μm diameter polystyrene sphere. (a) Whispering gallery modes are excited from a tangential or (b) a radial dipole source. The green vertical lines are the fundamental radial TE modes, and the red lines are the corresponding TM modes. The width of the bands accommodates the systematic uncertainty due to the finite grid size of FDTD.

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Fig. 6 Spatial distributions of the flux density over the collection region for several modes, S(r), integrated over a flux collection period of 1.0 ps, and projected onto the circular flux collection region. The modes considered are (a) TE40,1: 0.576 μm, (b) TE39,1: 0.588 μm, (c) TE38,1: 0.601 μm and (d) TE37,1: 0.615 μm. The axes are defined in the same way as in Fig. 1, for x-coordinate: 3.24 μm. Lighter color corresponds to larger magnitudes of S(r) · .

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Fig. 7 A comparison of the power spectra of 6 μm diameter microspheres with a tangential source (λ = 0.6 μm), (a) for flux collection regions at different distances Lflux from the centre of the sphere, and (b) for different flux region diameters Dflux. A resolution of Δx = 22 nm is used.

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The corresponding cavity Q-factors can be obtained by treating the simulation output as that of an experiment, and measuring the central wavelength position λ0, and the full-width at half-maximum, δλ [37]

Q=λ0δλ.
This formula will be used in estimating the simulation Q-factors in Table 2.

Tables Icon

Table 2. A summary of the Q-factors and wavelength positions, λ(μm), for the four most prominent WGM peaks, for each plot displayed in Fig. 4. The scenarios considered are: a surrounding medium of air or water with a tangential (Ey) or radial (Ex) dipole source. Due to finite collection time, a systematic uncertainty of up to 17% is expected in the Q-factors.

3. Analytic models

The simplest analytic WGM model that provides the positions of the TE and TM modes is particularly useful for identifying the mode numbers of the WGM peaks that occur in the FDTD spectrum. While this model identifies the resonance positions, which exactly match the results of Mie scattering theory, [40], this approach does not give the total electric and magnetic fields including the dipole source, and is unable to generate the profile of the spectrum.

The model is defined by obtaining the resonance condition as a function of wavelength for a dielectric sphere, and follows the formalism described in [38, 39]. Starting from a very general construction, a continuous electric field, oscillating with frequency ω, takes the form E(r) = E0(r) eiωt, for position r. This satisfies the wave equation:

××E0(r)k2n2(r)E0(r)=0,
for a refractive index n(r), and wave-number k = ω/c, where c is the speed of light in a vacuum.

The TE modes are obtained by separation of variables in spherical polar coordinates

E0=Sm(r)eimφkrXlm(θ),
Xlm(θ)=imsinθPlm(cosθ)e^θθPlm(cosθ)e^φ,
where the angular vector function Xlm(θ) is defined in terms of the associated Legendre polynomial Plm, and contains contributions in directions êθ and êφ. The function Sm(r) satisfies the second order differential equation
dSm(r)dr2+(k2n2(r)m(m+1)r2)Sm(r)=0.

Up to this point, the electric field simply takes the general form of a field in spherical polar coordinates, embedded in a dielectric medium. Now, the resonance condition may be obtained for a dielectric sphere, by specifying continuity across the boundary in the radial component of the electric field, Sm(r). For a refractive index of n1 inside a sphere of radius R (r ≡ |r| < R), and n2 outside, the radial function Sm(r) takes a Riccati-Bessel form, which can be expressed in terms of the spherical Bessel functions of the first kind, jm, and the spherical Hankel function, hm(1):

Sm(r)={z1rjm(z1r),r<RAmz2rhm(1)(z2r),r>R.
The use of the outgoing spherical Hankel function, hm(1)(z) takes into account leaky WGMs, which radiate energy outwards, and are described by quasi-modes [43] that are normalizable. The arguments z1,2rn1,2kr are the so-called size parameters, and Am are the coefficients defined through continuity at the boundary. The resonance condition for the TE modes of a sphere of radius R is thus [39]
n1z1R(m+1)jm(z1R)z1Rjm+1(z1R)jm(z1R)=n2z2R(m+1)ym(z2R)z2Rym+1(z2R)ym(z2R).

For TM modes, the form of E0 contains both angular and radial vector components (for êr as the unit vector in the radial direction)

E0=eimφk2n2(r)[1rTm(r)rYlm(θ)+1r2Tm(r)Zlm(θ)],
Ylm(θ)=e^r×Xlm(θ),
Zlm(θ)=l(l+1)Plm(cosθ)e^r,
where the function Tm(r) obeys the equation
dTm(r)dr22n(r)dn(r)drdTm(r)dr+(k2n2(r)m(m+1)r2)Tm(r)=0.
The solutions to Eq. (12) for Tm(r) take a similar form to those of Sm(r) above
Tm(r)={z1rjm(z1r),r<RBmz2rhm(1)(z2r),r>R,
where the coefficient Bm, analogously to Am, may be defined through enforcing continuity at the boundary. This leads to the resonance condition
1n1z1R(m+1)jm(z1R)z1jm+1(z1R)jm(z1R)=1n2z2R(m+1)ym(z2R)z2ym+1(z2R)ym(z2R).
By solving for the wave number k, a spectrum of independent TE and TM modes can be obtained, and compared to the FDTD simulation.

A more general type of analytic model, such as that developed by Chance, Prock and Silbey [26] and separately by Chew [25, 29, 30], is able to simulate the full electric and magnetic field behavior both inside and outside the resonator. This type of model will serve as an excellent benchmark comparison for the FDTD profile spectrum. Both models were shown to be equivalent formulations of spherical resonators excited by a dipole source [29], and obtain the same resonance positions as Mie scattering [40]. The simpler analytic models, however, are still useful at identifying the mode numbers of the peaks corresponding to WGMs in a microsphere. For a tangentially (Ey) or radially (Ex) oriented dipole source on the surface of a sphere of radius R and refractive index n1=ε1μ1, the power output equations take the following form (when normalized to the surrounding medium of index n2):

PEy/P0Ey=3ε13/2n12z1R2(ε2μ2)1/2m=1m(m+1)(2m+1)jm2(z1R)z1R2|Dm|2,
PEx/P0Ex=3ε13/2n14z1R2(ε2μ2)1/2m=1(2m+1)[|[z1Rjm(z1R)]z1RDm|2+μ1μ2ε1ε2jm2(z1R)|D˜m|2],
forDm=ε1jm(z1R)[z2Rhm(1)(z2R)]ε2hm(1)(z2R)[z1Rjm(z1R)],
D˜m=Dm(ε1,2μ1,2).
Although this analytic model is better suited to simulating scenarios that involve dipole sources at a variety of positions and alignments, the FDTD method is able to collect flux from any desired collection region (in the near or far field) for any length of time. This is an important point, as this collection scenario will affect the measured power spectrum profile. It is useful to be able to estimate the size of the effect due to collection region, and this will be investigated in the specific case of the angular distribution of the flux, in Section 4.

4. Results

An FDTD simulation of a polystyrene (n1 = 1.59) microsphere, with a diameter of 6 μm is carried out for a tangentially-oriented electric dipole source, emitting a Gaussian pulse with a central wavelength of 600 nm, and a width of 5 fs. Note that the pulse width is significantly narrower than the decay transition rate expected from a typical fluorescent source, such as Rhodamine dye, which is roughly 1–3 ns [44, 45]. In this work, the pulse is taken to be effectively instantaneous with respect to the phenomena being measured. A circular flux collection region with a diameter of 2.58 μm is placed a distance of 240 nm from the surface of the sphere in the x-direction, with its normal aligned along the same axis. Spectral information is then collected for wavelengths in the range 500–750 nm. The finite grid resolution entails an asymmetry in the Gaussian peak, which diminishes as the resolution increases.

A comparison of a variety of grid resolutions is shown in Fig. 2(a), using a total flux collection time of 0.6 ps (120 times the width of the pulse). For resolutions in each spatial direction of 22 – 33 nm, one finds that as the resolution decreases, the profile features of the spectrum do not alter significantly; there is only a small offset in the positions of the peaks, and the peak heights. The positions of the WGMs are determined from a flux collection of the frequencies, which are then converted to wavelengths. The temporal resolution is further interpolated to yield a value < 1 nm.

The systematic uncertainty in the resonance positions due to resolution can be quantified by tracking the positions of the WGM peaks. By examining the wavelength positions of the most prominent peaks, one can obtain a converged result to a chosen tolerance, as shown in Fig. 2(b). The convergence of the peak positions can be obtained by observing a systematic shift in the positions as the resolution is improved. For example, using a resolution of Δx = 25 nm, the most central peak has a wavelength position of 602.11 nm. By improving the resolution to Δx = 22 nm, the position becomes 601.35 nm. This yields an improvement in the position of the central peak of (602.11 – 601.35) = 0.76 nm, or 0.13%. This indicates a reasonable level of convergence of the peaks, as any further increase in resolution is expected to improve the result by 0.13% at most. Note that an even greater spectral resolution (∼ 0.03 nm) is required for the detection of nearby macromolecules by a microsphere [46]. This presents an issue in simulating the effect of detection using FDTD. However, FDTD is suitable for providing a realistic estimate of the WGM Q-factors under a variety of resonator scenarios, and for assessing their feasibility for new designs of biosensors.

By changing the length of time allowed for flux collection, one can also obtain important insights into the distribution of modes in the power spectrum. Figure 3(a) shows the resultant normalized power spectrum for a variety of flux collection times. A spatial resolution of Δx = 25 nm is chosen, and also a density of sampled wavelengths corresponding to Δλ = 0.62 nm. As the collection time increases, the WGM peaks become more prominent. Furthermore, double-peak structures seen at small collection times are no longer apparent for larger collection times, as the sampling of the full mode structure of the radiating cavity does not have sufficient time to be measured by the flux region. For the longer collection times, an enhancement in the normalized dipole power is observed (P/P0 > 1), at wavelengths coinciding with the dominant WGMs. This indicates that the power output of the source is enhanced at these wavelengths by the presence of the dielectric microsphere.

The behavior of the Q-factor as a function of collection time is shown in Fig. 3(b) for three central WGM peaks. For the peak at λ ∼ 0.615 μm, convergence is achieved beyond a collection time of 1.0 ps, with a variation of 3.4%. For the other two peaks, λ ∼ 0.601 μm and 0.589 μm, variations of 16.9% and 15.3% are measured, respectively. Note that, while peaks with lower Q-factors incur less variation with respect to collection time, peaks with higher Q-factors are more affected. However, an increase in collection time becomes increasingly computationally intensive, due to the asymptotic behavior in the Q-factor with respect to collection time [47]. Nevertheless, the uncertainty in the Q-factor due to the collection time can be quantified, and thus incorporated into the systematic uncertainty. This systematic effect will be considered when reporting the results in Table 2.

A comparison of the analytic model and the FDTD simulated spectrum for a microsphere is shown in Fig. 4 for tangentially and radially oriented dipole sources in both air and water media. The WGM positions from the analytic model are shown as vertical bands, with TEm,n modes in green and TMm,n modes in red, for azimuthal and radial mode numbers m and n, respectively. The width of each band indicates the estimate of the systematic uncertainty due to the finite grid spacing of FDTD. Specifically, the Mie scattering analytic model is used to estimate the shift in the WGM positions due to uncertainty in the sphere diameter, 6 μm ±Δx/2. In this case, the spatial resolution is held fixed at the value Δx = 22 nm. For the temporal resolution, the spectral density of 800 points yields an uncertainty of (750 − 500)nm/800 = 0.31 nm. The two uncertainties are added in quadrature. The small offset of each peak from the position expected from the analytic model may be due to mixing of nearby modes, which interact with each other to shift the peaks away from their central positions. The TE and TM modes cannot be completely decoupled due to the spherical symmetry, and contributions from both TE and TM modes are expected in the spectrum, in addition to non-whispering gallery radiation modes.

For a tangential electric dipole (Fig. 4(a)), one expects that the dominant WGMs excited are the lowest-order radial TE modes, since the electric field of each of the TM modes contains a radial component (see Section 3), which does not couple strongly to the tangential source. It is found that the dominant peaks have a free spectral range (FSR) consistent with a radial mode number of n = 1. Note that the fundamental radial modes (n = 0) cannot be resolved for this index contrast at this finite grid size, since they are known to exhibit extremely large Q-factors experimentally [48] and in the Chew model [30]. In the case of the radial electric dipole (Fig. 4(b)), the dominant peaks exhibit an FSR that matches the n = 1 and n = 2 TM modes. In this case, there appears to be very little contribution from the TE modes.

For a lower index contrast scenario, such as polystyrene microspheres in a surrounding medium of water (Figs. 4(c) and 4(d)), the WGM peaks are broadened, and there is a reduced density of modes at wavelengths in the range 500–750 nm. As a result, the peaks observed in the FDTD simulation correspond to the fundamental radial TE/TM modes, for a tangential/radial dipole source, respectively.

Comparisons of the Chew model, the Mie scattering analytic model and the FDTD simulations are shown in Fig. 5, for a surrounding medium of water. In Fig. 5(a), a tangential dipole source is used, and the fundamental radial TE modes from the Mie scattering model (green vertical lines) exactly match the peaks of the Chew model (dashed blue line), as expected by construction [30]. The peaks of the FDTD simulation also correspond to these TE modes, and the FSR matches that of the Chew model, with a small peak offset due to the finite grid resolution. Figure 5(b) shows the result for a radial dipole. The peaks of both the Chew model and the FDTD simulation correspond to the fundamental radial TM modes.

The simulation can also provide information about the angular distribution of the flux, both directly and indirectly. Since the FDTD simulation records the electromagnetic field values, the flux density, S(r), may be projected onto the circular region of the zy plane (with normal vector ) for any wavelength value. An analysis of this type is helpful for visualizing the distribution, distinguishing the azimuthal and radial modes at different time slices, and identifying transient resonant features. By integrating the flux over a collection time of 1.0 ps, the distribution over the collection plane indicates the angular dependence of the mode at a particular wavelength. As an example, Fig. 6 shows the flux distribution over the collection region, for a tangential dipole source, and a surrounding medium of air. The four panels display the four most prominent WGM peaks occurring in that wavelength region, corresponding to values of 0.576, 0.588, 0.601 and 0.615 μm, respectively. It is apparent that the modes at wavelengths of 0.601 μm (TE38,1) and 0.615 μm (TE37,1) exhibit a flux distribution with peaks spread out in the flux measuring region over a comparatively wide angle. The total power output for these modes is likely to vary significantly if the diameter of the region is reduced. In contrast, the mode occurring at 0.588 μm (TE39,1) has a more focused concentration of flux in the centre of the measuring region, corresponding to a narrower angular distribution of flux. The power output for this mode will be consistent over a wider range of flux region diameters.

As a consequence of the nontrivial angular distributions of the modes, changes in the power spectrum are also detected. Figure 7 shows the spectrum for the 6 μm sphere for flux planes at different distances from the sphere (Fig. 7(a)), and different diameters (Fig. 7(b)). In each case, a tangential electric dipole is used, and the collection time is held fixed at 1.0 ps. By comparing the spectra from several flux collection plane sizes and positions, changes in the peak heights are observed, indicating the different angular behavior of the modes. Changes in the distance of the flux region, Lflux, result in a similar mode structure, but the overall power output, especially for the dominant WGM peaks, changes as the flux is sampled differently in each region. Alterations in the diameter of the flux region results in more noticeable changes to the mode heights, demonstrating the highly angular-dependent nature of the relative contribution of each mode to the total flux.

These scenarios indicate that the FDTD simulation package represents a promising development toward a robust, customizable, predictive toolkit for simulating the whispering gallery modes of microspherical resonators. In this investigation, the FDTD tool has already provided guidance for future designs of resonators, which can be adapted for biosensing applications. For example, in Fig. 4, dominance of TE or TM modes is dependent on the alignment of the dipole source on the surface of the sphere. Microsphere sensors may be able to detect the orientation of an external biomolecule by recording the relative coupling strengths of TE to TM modes.

5. Conclusion

This work presents the first easily customizable approach for simulating whispering gallery modes of microspheres, which is based on the FDTD method. The tool is easily customizable, and especially suited to investigating a wide variety of resonator scenarios. The FDTD approach is able to simulate changes to the power spectrum by measuring flux in the near-field region for any length of time, using a circular flux collection plane. The comprehensive inclusion of all modes allows for more realistic simulations than is feasible for typical analytic models.

In addition, different methods of mode excitation of a microsphere can be investigated, such as dipole source locations and alignments. Changes in the peak positions of the modes due to discretization effects can be quantified, and absorbed into the systematic uncertainty.

In practice, one can scan over a variety of parameters to gauge the effect on the profile of the collected power spectrum. Consequently, a wavelength and resonator combination can be sought systematically, and optical attributes of particular interest may be emphasized or suppressed. The optimization of the coupling efficiency to certain mode channels allows one to preselect desired resonator properties in a cost-effective way, leading to new design solutions that might not otherwise have been found.

As a first example, the angular behavior of the energy flux was investigated for several sizes and distances of the collection region, leading to alterations in the coupling efficiencies to different modes. This is observed in the profile of the power spectrum. The magnitude of the flux density projected onto the collection disk was also mapped out for a range of collection times, allowing one to distinguish the distribution of the modes at different collection times.

The new simulation results were compared and contrasted to two different analytic models. The analytic models rely on assumptions of a perfectly smooth dielectric, with the total radiation output measured in the far-field region. It was found that the peak positions from the models matched the FDTD simulations well.

The computational toolkit presented in this paper represents the first step in establishing a realistic and easy-to-use simulator, which is important for facilitating a cost-effective approach to designing tailored optical resonators for biosensing.

Acknowledgments

The authors acknowledge the support of the ARC Georgina Sweet Laureate Fellowship (T. M. M.), and the ARC Centre for Nanoscale BioPhotonics. The authors also acknowledge eResearchSA for the use of supercomputing resources.

References and links

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2. F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002). [CrossRef]  

3. F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: Label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008). [CrossRef]   [PubMed]  

4. R. W. Boyd and J. E. Heebner, “Sensitive disk resonator photonic biosensor,” Appl. Opt. 40, 5742–5747 (2001). [CrossRef]  

5. S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett. 28, 272–274 (2003). [CrossRef]   [PubMed]  

6. A. Ksendzov and Y. Lin, “Integrated optics ring-resonator sensors for protein detection,” Opt. Lett. 30, 3344–3346 (2005). [CrossRef]  

7. J. D. Suter, I. M. White, H. Zhu, H. Shi, C. W. Caldwell, and X. Fan, “Label-free quantitative DNA detection using the liquid core optical ring resonator,” Biosens. Bioelectron. 23, 1003–1009 (2008). [CrossRef]  

8. F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics 1(3–4) 267–291 (2012). [CrossRef]  

9. A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783–787 (2007). [CrossRef]   [PubMed]  

10. M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9, 933–939 (2014). [CrossRef]   [PubMed]  

11. W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Generation of near-infrared frequency combs from a MgF2 whispering gallery mode resonator,” Opt. Lett. 36, 2290–2292 (2011). [CrossRef]   [PubMed]  

12. A. Schliesser and T. J. Kippenberg, “Chapter 5 - cavity optomechanics with whispering-gallery mode optical micro-resonators,” in “Advances In Atomic, Molecular, and Optical Physics,”, vol. 58 of Advances In Atomic, Molecular, and Optical Physics, E. A. Paul Berman and C. Lin, eds. (Academic Press, 2010), pp. 207–323. [CrossRef]  

13. D. L. McAuslan, D. Korystov, and J. J. Longdell, “Coherent spectroscopy of rare-earth-metal-ion-doped whispering-gallery-mode resonators,” Phys. Rev. A 83, 063847 (2011). [CrossRef]  

14. J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, “Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper,” Opt. Lett. 22, 1129–1131 (1997). [CrossRef]   [PubMed]  

15. M. Hossein-Zadeh and K. J. Vahala, “Fiber-taper coupling to whispering-gallery modes of fluidic resonators embedded in a liquid medium,” Opt. Express 14, 10800–10810 (2006). [CrossRef]   [PubMed]  

16. D. Xiao-Wei, L. Shao-Hua, F. Su-Chun, X. Ou, and J. Shui-Sheng, “All-fibre micro-ring resonator based on tapered microfibre,” Chin. Phys. B 17, 1029 (2008). [CrossRef]  

17. A. Boleininger, T. Lake, S. Hami, and C. Vallance, “Whispering gallery modes in standard optical fibres for fibre profiling measurements and sensing of unlabelled chemical species,” Sensors 10, 1765–1781 (2010). [CrossRef]   [PubMed]  

18. F. Monifi, S. Odemir, J. Friedlein, and L. Yang, “Encapsulation of a fiber taper coupled microtoroid resonator in a polymer matrix,” IEEE Photon. Technol. Lett. 25, 1458–1461 (2013). [CrossRef]  

19. A. François, K. J. Rowland, V. S. Afshar, M. R. Henderson, and T. M. Monro, “Enhancing the radiation efficiency of dye doped whispering gallery mode microresonators,” Opt. Express 21, 22566–22577 (2013). [CrossRef]   [PubMed]  

20. B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457, 455–458 (2009). [CrossRef]   [PubMed]  

21. S. I. Shopova, G. Farca, A. Naweed, and A. T. Rosenberger, “Whispering-gallery-mode microlaser consisting of a HgTe-quantum-dot-coated microsphere,” in Proceedings of Frontiers in Optics (2003), p. TuT4.

22. M. Fujii, M. Haraguchi, T. Okamoto, and M. Fukui, “Characteristics of whispering gallery modes in single dielectric spheroid excited by gaussian beam,” Jpn. J. Appl. Phys. 44, 4948 (2005). [CrossRef]  

23. M. Fujii, M. Haraguchi, T. Okamoto, and M. Fukui, “Characteristics of whispering gallery modes in single spheroid: Dependence on the direction of incident light,” J. Korean Phys. Soc. 47, S38–S42 (2005).

24. H. Quan and Z. Guo, “Simulation of whispering-gallery-mode resonance shifts for optical miniature biosensors,” J. Quant. Spectrosc. Radiat. Trans. 93, 231–243 (2005). [CrossRef]  

25. H. Chew, P. J. McNulty, and M. Kerker, “Model for raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976). [CrossRef]  

26. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 65 (1978).

27. J. Gersten and A. Nitzan, “Electromagnetic theory of enhanced Raman scattering by molecules adsorbed on rough surfaces,” J. Chem. Phys. 73, 3023–3037 (1980). [CrossRef]  

28. J. Gersten and A. Nitzan, “Spectroscopic properties of molecules interacting with small dielectric particles,” J. Chem. Phys. 75, 1139–1152 (1981). [CrossRef]  

29. H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987). [CrossRef]  

30. H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988). [CrossRef]   [PubMed]  

31. R. Ruppin, “Decay of an excited molecule near a small metal sphere,” J. Chem. Phys. 76, 1681–1684 (1982). [CrossRef]  

32. M. K. Schmidt, R. Esteban, J. J. Sáenz, I. Suárez-Lacalle, S. Mackowski, and J. Aizpurua, “Dielectric antennas -a suitable platform for controlling magnetic dipolar emission,” Opt. Express 20, 13636–13650 (2012). [CrossRef]   [PubMed]  

33. V S. Afshar, M. R. Henderson, A. D. Greentree, B. C. Gibson, and T. M. Monro, “Self-formed cavity quantum electrodynamics in coupled dipole cylindrical-waveguide systems,” Opt. Express 22, 11301–11311 (2014). [CrossRef]  

34. A. Taflove, Computational Electrodynamics: The Finite - Difference Time - Domain Method, Antennas and Propagation Library (Artech House, Inc., 1995).

35. P. S. Kuo, W. Fang, and G. S. Solomon, “4̄-quasi-phase-matched interactions in GaAs microdisk cavities,” Opt. Lett. 34, 3580 (2009). [CrossRef]   [PubMed]  

36. B. E. Little, J.-P. Laine, H. A. Haus, and L. Fellow, “Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators,” J. Lightwave Technol. 17, 704 (1999). [CrossRef]  

37. K. Vahala, Optical Microcavities, Advanced series in applied physics (World Scientific, 2004).

38. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10, 343–352 (1993). [CrossRef]  

39. I. Teraoka and S. Arnold, “Theory of resonance shifts in TE and TM whispering gallery modes by nonradial perturbations for sensing applications,” J. Opt. Soc. Am. B 23, 1381–1389 (2006). [CrossRef]  

40. C. F. Bohren and D. R. Huffman, Absorption and Scattering by a Sphere (Wiley-VCH Verlag GmbH, 2007), pp. 82–129.

41. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010). [CrossRef]  

42. T. Søndergaard and B. Tromborg, “General theory for spontaneous emission in active dielectric microstructures: example of a fiber amplifier,” Phys. Rev. A 64, 033812 (2001). [CrossRef]  

43. H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990). [CrossRef]   [PubMed]  

44. M. D. Barnes, W. B. Whitten, S. Arnold, and J. M. Ramsey, “Homogeneous linewidths of rhodamine 6G at room temperature from cavity-enhanced spontaneous emission rates,” J. Chem. Phys. 97, 7842–7845 (1992). [CrossRef]  

45. K. Kolmakov, V. N. Belov, J. Bierwagen, C. Ringemann, V. Muller, C. Eggeling, and S. W. Hell, “Red-emitting rhodamine dyes for fluorescence microscopy and nanoscopy,” Chem. Eur. J. 16, 158–166 (2010). [CrossRef]  

46. M. Himmelhaus, S. Krishnamoorthy, and A. François, “Optical sensors based on whispering gallery modes in fluorescent microbeads: response to specific interactions,” Sensors 10, 6257–6274 (2010). [CrossRef]   [PubMed]  

47. C. Marki, and S. D. E. E. P. University of California, Design and Optimization of Bidirectional and Optical Logic Systems in the Presence of Noise (University of CaliforniaSan Diego, 2007).

48. A. François, K. J. Rowland, and T. M. Monro, “Highly efficient excitation and detection of whispering gallery modes in a dye-doped microsphere using a microstructured optical fiber,” Appl. Phys. Lett. 99, 141111 (2011). [CrossRef]  

References

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  1. V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Lett. 13, 3347–3351 (2013).
    [Crossref] [PubMed]
  2. F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
    [Crossref]
  3. F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: Label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
    [Crossref] [PubMed]
  4. R. W. Boyd and J. E. Heebner, “Sensitive disk resonator photonic biosensor,” Appl. Opt. 40, 5742–5747 (2001).
    [Crossref]
  5. S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett. 28, 272–274 (2003).
    [Crossref] [PubMed]
  6. A. Ksendzov and Y. Lin, “Integrated optics ring-resonator sensors for protein detection,” Opt. Lett. 30, 3344–3346 (2005).
    [Crossref]
  7. J. D. Suter, I. M. White, H. Zhu, H. Shi, C. W. Caldwell, and X. Fan, “Label-free quantitative DNA detection using the liquid core optical ring resonator,” Biosens. Bioelectron. 23, 1003–1009 (2008).
    [Crossref]
  8. F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics 1(3–4) 267–291 (2012).
    [Crossref]
  9. A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783–787 (2007).
    [Crossref] [PubMed]
  10. M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9, 933–939 (2014).
    [Crossref] [PubMed]
  11. W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Generation of near-infrared frequency combs from a MgF2 whispering gallery mode resonator,” Opt. Lett. 36, 2290–2292 (2011).
    [Crossref] [PubMed]
  12. A. Schliesser and T. J. Kippenberg, “Chapter 5 - cavity optomechanics with whispering-gallery mode optical micro-resonators,” in “Advances In Atomic, Molecular, and Optical Physics,”, vol. 58 of Advances In Atomic, Molecular, and Optical Physics, E. A. Paul Berman and C. Lin, eds. (Academic Press, 2010), pp. 207–323.
    [Crossref]
  13. D. L. McAuslan, D. Korystov, and J. J. Longdell, “Coherent spectroscopy of rare-earth-metal-ion-doped whispering-gallery-mode resonators,” Phys. Rev. A 83, 063847 (2011).
    [Crossref]
  14. J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, “Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper,” Opt. Lett. 22, 1129–1131 (1997).
    [Crossref] [PubMed]
  15. M. Hossein-Zadeh and K. J. Vahala, “Fiber-taper coupling to whispering-gallery modes of fluidic resonators embedded in a liquid medium,” Opt. Express 14, 10800–10810 (2006).
    [Crossref] [PubMed]
  16. D. Xiao-Wei, L. Shao-Hua, F. Su-Chun, X. Ou, and J. Shui-Sheng, “All-fibre micro-ring resonator based on tapered microfibre,” Chin. Phys. B 17, 1029 (2008).
    [Crossref]
  17. A. Boleininger, T. Lake, S. Hami, and C. Vallance, “Whispering gallery modes in standard optical fibres for fibre profiling measurements and sensing of unlabelled chemical species,” Sensors 10, 1765–1781 (2010).
    [Crossref] [PubMed]
  18. F. Monifi, S. Odemir, J. Friedlein, and L. Yang, “Encapsulation of a fiber taper coupled microtoroid resonator in a polymer matrix,” IEEE Photon. Technol. Lett. 25, 1458–1461 (2013).
    [Crossref]
  19. A. François, K. J. Rowland, V. S. Afshar, M. R. Henderson, and T. M. Monro, “Enhancing the radiation efficiency of dye doped whispering gallery mode microresonators,” Opt. Express 21, 22566–22577 (2013).
    [Crossref] [PubMed]
  20. B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457, 455–458 (2009).
    [Crossref] [PubMed]
  21. S. I. Shopova, G. Farca, A. Naweed, and A. T. Rosenberger, “Whispering-gallery-mode microlaser consisting of a HgTe-quantum-dot-coated microsphere,” in Proceedings of Frontiers in Optics (2003), p. TuT4.
  22. M. Fujii, M. Haraguchi, T. Okamoto, and M. Fukui, “Characteristics of whispering gallery modes in single dielectric spheroid excited by gaussian beam,” Jpn. J. Appl. Phys. 44, 4948 (2005).
    [Crossref]
  23. M. Fujii, M. Haraguchi, T. Okamoto, and M. Fukui, “Characteristics of whispering gallery modes in single spheroid: Dependence on the direction of incident light,” J. Korean Phys. Soc. 47, S38–S42 (2005).
  24. H. Quan and Z. Guo, “Simulation of whispering-gallery-mode resonance shifts for optical miniature biosensors,” J. Quant. Spectrosc. Radiat. Trans. 93, 231–243 (2005).
    [Crossref]
  25. H. Chew, P. J. McNulty, and M. Kerker, “Model for raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
    [Crossref]
  26. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 65 (1978).
  27. J. Gersten and A. Nitzan, “Electromagnetic theory of enhanced Raman scattering by molecules adsorbed on rough surfaces,” J. Chem. Phys. 73, 3023–3037 (1980).
    [Crossref]
  28. J. Gersten and A. Nitzan, “Spectroscopic properties of molecules interacting with small dielectric particles,” J. Chem. Phys. 75, 1139–1152 (1981).
    [Crossref]
  29. H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987).
    [Crossref]
  30. H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
    [Crossref] [PubMed]
  31. R. Ruppin, “Decay of an excited molecule near a small metal sphere,” J. Chem. Phys. 76, 1681–1684 (1982).
    [Crossref]
  32. M. K. Schmidt, R. Esteban, J. J. Sáenz, I. Suárez-Lacalle, S. Mackowski, and J. Aizpurua, “Dielectric antennas -a suitable platform for controlling magnetic dipolar emission,” Opt. Express 20, 13636–13650 (2012).
    [Crossref] [PubMed]
  33. V S. Afshar, M. R. Henderson, A. D. Greentree, B. C. Gibson, and T. M. Monro, “Self-formed cavity quantum electrodynamics in coupled dipole cylindrical-waveguide systems,” Opt. Express 22, 11301–11311 (2014).
    [Crossref]
  34. A. Taflove, Computational Electrodynamics: The Finite - Difference Time - Domain Method, Antennas and Propagation Library (Artech House, Inc., 1995).
  35. P. S. Kuo, W. Fang, and G. S. Solomon, “4̄-quasi-phase-matched interactions in GaAs microdisk cavities,” Opt. Lett. 34, 3580 (2009).
    [Crossref] [PubMed]
  36. B. E. Little, J.-P. Laine, H. A. Haus, and L. Fellow, “Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators,” J. Lightwave Technol. 17, 704 (1999).
    [Crossref]
  37. K. Vahala, Optical Microcavities, Advanced series in applied physics (World Scientific, 2004).
  38. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10, 343–352 (1993).
    [Crossref]
  39. I. Teraoka and S. Arnold, “Theory of resonance shifts in TE and TM whispering gallery modes by nonradial perturbations for sensing applications,” J. Opt. Soc. Am. B 23, 1381–1389 (2006).
    [Crossref]
  40. C. F. Bohren and D. R. Huffman, Absorption and Scattering by a Sphere (Wiley-VCH Verlag GmbH, 2007), pp. 82–129.
  41. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
    [Crossref]
  42. T. Søndergaard and B. Tromborg, “General theory for spontaneous emission in active dielectric microstructures: example of a fiber amplifier,” Phys. Rev. A 64, 033812 (2001).
    [Crossref]
  43. H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
    [Crossref] [PubMed]
  44. M. D. Barnes, W. B. Whitten, S. Arnold, and J. M. Ramsey, “Homogeneous linewidths of rhodamine 6G at room temperature from cavity-enhanced spontaneous emission rates,” J. Chem. Phys. 97, 7842–7845 (1992).
    [Crossref]
  45. K. Kolmakov, V. N. Belov, J. Bierwagen, C. Ringemann, V. Muller, C. Eggeling, and S. W. Hell, “Red-emitting rhodamine dyes for fluorescence microscopy and nanoscopy,” Chem. Eur. J. 16, 158–166 (2010).
    [Crossref]
  46. M. Himmelhaus, S. Krishnamoorthy, and A. François, “Optical sensors based on whispering gallery modes in fluorescent microbeads: response to specific interactions,” Sensors 10, 6257–6274 (2010).
    [Crossref] [PubMed]
  47. C. Marki and S. D. E. E. P. University of California, Design and Optimization of Bidirectional and Optical Logic Systems in the Presence of Noise (University of CaliforniaSan Diego, 2007).
  48. A. François, K. J. Rowland, and T. M. Monro, “Highly efficient excitation and detection of whispering gallery modes in a dye-doped microsphere using a microstructured optical fiber,” Appl. Phys. Lett. 99, 141111 (2011).
    [Crossref]

2014 (2)

M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9, 933–939 (2014).
[Crossref] [PubMed]

V S. Afshar, M. R. Henderson, A. D. Greentree, B. C. Gibson, and T. M. Monro, “Self-formed cavity quantum electrodynamics in coupled dipole cylindrical-waveguide systems,” Opt. Express 22, 11301–11311 (2014).
[Crossref]

2013 (3)

F. Monifi, S. Odemir, J. Friedlein, and L. Yang, “Encapsulation of a fiber taper coupled microtoroid resonator in a polymer matrix,” IEEE Photon. Technol. Lett. 25, 1458–1461 (2013).
[Crossref]

A. François, K. J. Rowland, V. S. Afshar, M. R. Henderson, and T. M. Monro, “Enhancing the radiation efficiency of dye doped whispering gallery mode microresonators,” Opt. Express 21, 22566–22577 (2013).
[Crossref] [PubMed]

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Lett. 13, 3347–3351 (2013).
[Crossref] [PubMed]

2012 (2)

F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics 1(3–4) 267–291 (2012).
[Crossref]

M. K. Schmidt, R. Esteban, J. J. Sáenz, I. Suárez-Lacalle, S. Mackowski, and J. Aizpurua, “Dielectric antennas -a suitable platform for controlling magnetic dipolar emission,” Opt. Express 20, 13636–13650 (2012).
[Crossref] [PubMed]

2011 (3)

W. Liang, A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Generation of near-infrared frequency combs from a MgF2 whispering gallery mode resonator,” Opt. Lett. 36, 2290–2292 (2011).
[Crossref] [PubMed]

D. L. McAuslan, D. Korystov, and J. J. Longdell, “Coherent spectroscopy of rare-earth-metal-ion-doped whispering-gallery-mode resonators,” Phys. Rev. A 83, 063847 (2011).
[Crossref]

A. François, K. J. Rowland, and T. M. Monro, “Highly efficient excitation and detection of whispering gallery modes in a dye-doped microsphere using a microstructured optical fiber,” Appl. Phys. Lett. 99, 141111 (2011).
[Crossref]

2010 (4)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[Crossref]

K. Kolmakov, V. N. Belov, J. Bierwagen, C. Ringemann, V. Muller, C. Eggeling, and S. W. Hell, “Red-emitting rhodamine dyes for fluorescence microscopy and nanoscopy,” Chem. Eur. J. 16, 158–166 (2010).
[Crossref]

M. Himmelhaus, S. Krishnamoorthy, and A. François, “Optical sensors based on whispering gallery modes in fluorescent microbeads: response to specific interactions,” Sensors 10, 6257–6274 (2010).
[Crossref] [PubMed]

A. Boleininger, T. Lake, S. Hami, and C. Vallance, “Whispering gallery modes in standard optical fibres for fibre profiling measurements and sensing of unlabelled chemical species,” Sensors 10, 1765–1781 (2010).
[Crossref] [PubMed]

2009 (2)

P. S. Kuo, W. Fang, and G. S. Solomon, “4̄-quasi-phase-matched interactions in GaAs microdisk cavities,” Opt. Lett. 34, 3580 (2009).
[Crossref] [PubMed]

B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457, 455–458 (2009).
[Crossref] [PubMed]

2008 (3)

D. Xiao-Wei, L. Shao-Hua, F. Su-Chun, X. Ou, and J. Shui-Sheng, “All-fibre micro-ring resonator based on tapered microfibre,” Chin. Phys. B 17, 1029 (2008).
[Crossref]

J. D. Suter, I. M. White, H. Zhu, H. Shi, C. W. Caldwell, and X. Fan, “Label-free quantitative DNA detection using the liquid core optical ring resonator,” Biosens. Bioelectron. 23, 1003–1009 (2008).
[Crossref]

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: Label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[Crossref] [PubMed]

2007 (1)

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783–787 (2007).
[Crossref] [PubMed]

2006 (2)

2005 (4)

M. Fujii, M. Haraguchi, T. Okamoto, and M. Fukui, “Characteristics of whispering gallery modes in single dielectric spheroid excited by gaussian beam,” Jpn. J. Appl. Phys. 44, 4948 (2005).
[Crossref]

M. Fujii, M. Haraguchi, T. Okamoto, and M. Fukui, “Characteristics of whispering gallery modes in single spheroid: Dependence on the direction of incident light,” J. Korean Phys. Soc. 47, S38–S42 (2005).

H. Quan and Z. Guo, “Simulation of whispering-gallery-mode resonance shifts for optical miniature biosensors,” J. Quant. Spectrosc. Radiat. Trans. 93, 231–243 (2005).
[Crossref]

A. Ksendzov and Y. Lin, “Integrated optics ring-resonator sensors for protein detection,” Opt. Lett. 30, 3344–3346 (2005).
[Crossref]

2003 (1)

2002 (1)

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[Crossref]

2001 (2)

R. W. Boyd and J. E. Heebner, “Sensitive disk resonator photonic biosensor,” Appl. Opt. 40, 5742–5747 (2001).
[Crossref]

T. Søndergaard and B. Tromborg, “General theory for spontaneous emission in active dielectric microstructures: example of a fiber amplifier,” Phys. Rev. A 64, 033812 (2001).
[Crossref]

1999 (1)

1997 (1)

1993 (1)

1992 (1)

M. D. Barnes, W. B. Whitten, S. Arnold, and J. M. Ramsey, “Homogeneous linewidths of rhodamine 6G at room temperature from cavity-enhanced spontaneous emission rates,” J. Chem. Phys. 97, 7842–7845 (1992).
[Crossref]

1990 (1)

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[Crossref] [PubMed]

1988 (1)

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[Crossref] [PubMed]

1987 (1)

H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987).
[Crossref]

1982 (1)

R. Ruppin, “Decay of an excited molecule near a small metal sphere,” J. Chem. Phys. 76, 1681–1684 (1982).
[Crossref]

1981 (1)

J. Gersten and A. Nitzan, “Spectroscopic properties of molecules interacting with small dielectric particles,” J. Chem. Phys. 75, 1139–1152 (1981).
[Crossref]

1980 (1)

J. Gersten and A. Nitzan, “Electromagnetic theory of enhanced Raman scattering by molecules adsorbed on rough surfaces,” J. Chem. Phys. 73, 3023–3037 (1980).
[Crossref]

1978 (1)

R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 65 (1978).

1976 (1)

H. Chew, P. J. McNulty, and M. Kerker, “Model for raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[Crossref]

Aizpurua, J.

Armani, A. M.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783–787 (2007).
[Crossref] [PubMed]

Arnold, S.

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Lett. 13, 3347–3351 (2013).
[Crossref] [PubMed]

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: Label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[Crossref] [PubMed]

I. Teraoka and S. Arnold, “Theory of resonance shifts in TE and TM whispering gallery modes by nonradial perturbations for sensing applications,” J. Opt. Soc. Am. B 23, 1381–1389 (2006).
[Crossref]

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett. 28, 272–274 (2003).
[Crossref] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[Crossref]

M. D. Barnes, W. B. Whitten, S. Arnold, and J. M. Ramsey, “Homogeneous linewidths of rhodamine 6G at room temperature from cavity-enhanced spontaneous emission rates,” J. Chem. Phys. 97, 7842–7845 (1992).
[Crossref]

Baaske, M. D.

M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9, 933–939 (2014).
[Crossref] [PubMed]

Barber, P. W.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[Crossref] [PubMed]

Barbre, C.

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Lett. 13, 3347–3351 (2013).
[Crossref] [PubMed]

Barnes, M. D.

M. D. Barnes, W. B. Whitten, S. Arnold, and J. M. Ramsey, “Homogeneous linewidths of rhodamine 6G at room temperature from cavity-enhanced spontaneous emission rates,” J. Chem. Phys. 97, 7842–7845 (1992).
[Crossref]

Belov, V. N.

K. Kolmakov, V. N. Belov, J. Bierwagen, C. Ringemann, V. Muller, C. Eggeling, and S. W. Hell, “Red-emitting rhodamine dyes for fluorescence microscopy and nanoscopy,” Chem. Eur. J. 16, 158–166 (2010).
[Crossref]

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[Crossref]

Bierwagen, J.

K. Kolmakov, V. N. Belov, J. Bierwagen, C. Ringemann, V. Muller, C. Eggeling, and S. W. Hell, “Red-emitting rhodamine dyes for fluorescence microscopy and nanoscopy,” Chem. Eur. J. 16, 158–166 (2010).
[Crossref]

Birks, T. A.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering by a Sphere (Wiley-VCH Verlag GmbH, 2007), pp. 82–129.

Boleininger, A.

A. Boleininger, T. Lake, S. Hami, and C. Vallance, “Whispering gallery modes in standard optical fibres for fibre profiling measurements and sensing of unlabelled chemical species,” Sensors 10, 1765–1781 (2010).
[Crossref] [PubMed]

Boyd, R. W.

Braun, D.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[Crossref]

Caldwell, C. W.

J. D. Suter, I. M. White, H. Zhu, H. Shi, C. W. Caldwell, and X. Fan, “Label-free quantitative DNA detection using the liquid core optical ring resonator,” Biosens. Bioelectron. 23, 1003–1009 (2008).
[Crossref]

Chance, R.

R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 65 (1978).

Cheung, G.

Chew, H.

H. Chew, “Radiation and lifetimes of atoms inside dielectric particles,” Phys. Rev. A 38, 3410–3416 (1988).
[Crossref] [PubMed]

H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1987).
[Crossref]

H. Chew, P. J. McNulty, and M. Kerker, “Model for raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[Crossref]

Dantham, V. R.

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Lett. 13, 3347–3351 (2013).
[Crossref] [PubMed]

Eggeling, C.

K. Kolmakov, V. N. Belov, J. Bierwagen, C. Ringemann, V. Muller, C. Eggeling, and S. W. Hell, “Red-emitting rhodamine dyes for fluorescence microscopy and nanoscopy,” Chem. Eur. J. 16, 158–166 (2010).
[Crossref]

Esteban, R.

Fan, X.

J. D. Suter, I. M. White, H. Zhu, H. Shi, C. W. Caldwell, and X. Fan, “Label-free quantitative DNA detection using the liquid core optical ring resonator,” Biosens. Bioelectron. 23, 1003–1009 (2008).
[Crossref]

Fang, W.

Farca, G.

S. I. Shopova, G. Farca, A. Naweed, and A. T. Rosenberger, “Whispering-gallery-mode microlaser consisting of a HgTe-quantum-dot-coated microsphere,” in Proceedings of Frontiers in Optics (2003), p. TuT4.

Fellow, L.

Flagan, R. C.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783–787 (2007).
[Crossref] [PubMed]

Foreman, M. R.

M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9, 933–939 (2014).
[Crossref] [PubMed]

François, A.

A. François, K. J. Rowland, V. S. Afshar, M. R. Henderson, and T. M. Monro, “Enhancing the radiation efficiency of dye doped whispering gallery mode microresonators,” Opt. Express 21, 22566–22577 (2013).
[Crossref] [PubMed]

A. François, K. J. Rowland, and T. M. Monro, “Highly efficient excitation and detection of whispering gallery modes in a dye-doped microsphere using a microstructured optical fiber,” Appl. Phys. Lett. 99, 141111 (2011).
[Crossref]

M. Himmelhaus, S. Krishnamoorthy, and A. François, “Optical sensors based on whispering gallery modes in fluorescent microbeads: response to specific interactions,” Sensors 10, 6257–6274 (2010).
[Crossref] [PubMed]

Fraser, S. E.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783–787 (2007).
[Crossref] [PubMed]

Friedlein, J.

F. Monifi, S. Odemir, J. Friedlein, and L. Yang, “Encapsulation of a fiber taper coupled microtoroid resonator in a polymer matrix,” IEEE Photon. Technol. Lett. 25, 1458–1461 (2013).
[Crossref]

Fujii, M.

M. Fujii, M. Haraguchi, T. Okamoto, and M. Fukui, “Characteristics of whispering gallery modes in single dielectric spheroid excited by gaussian beam,” Jpn. J. Appl. Phys. 44, 4948 (2005).
[Crossref]

M. Fujii, M. Haraguchi, T. Okamoto, and M. Fukui, “Characteristics of whispering gallery modes in single spheroid: Dependence on the direction of incident light,” J. Korean Phys. Soc. 47, S38–S42 (2005).

Fukui, M.

M. Fujii, M. Haraguchi, T. Okamoto, and M. Fukui, “Characteristics of whispering gallery modes in single spheroid: Dependence on the direction of incident light,” J. Korean Phys. Soc. 47, S38–S42 (2005).

M. Fujii, M. Haraguchi, T. Okamoto, and M. Fukui, “Characteristics of whispering gallery modes in single dielectric spheroid excited by gaussian beam,” Jpn. J. Appl. Phys. 44, 4948 (2005).
[Crossref]

Gersten, J.

J. Gersten and A. Nitzan, “Spectroscopic properties of molecules interacting with small dielectric particles,” J. Chem. Phys. 75, 1139–1152 (1981).
[Crossref]

J. Gersten and A. Nitzan, “Electromagnetic theory of enhanced Raman scattering by molecules adsorbed on rough surfaces,” J. Chem. Phys. 73, 3023–3037 (1980).
[Crossref]

Gibson, B. C.

Greentree, A. D.

Guo, Z.

H. Quan and Z. Guo, “Simulation of whispering-gallery-mode resonance shifts for optical miniature biosensors,” J. Quant. Spectrosc. Radiat. Trans. 93, 231–243 (2005).
[Crossref]

Hami, S.

A. Boleininger, T. Lake, S. Hami, and C. Vallance, “Whispering gallery modes in standard optical fibres for fibre profiling measurements and sensing of unlabelled chemical species,” Sensors 10, 1765–1781 (2010).
[Crossref] [PubMed]

Haraguchi, M.

M. Fujii, M. Haraguchi, T. Okamoto, and M. Fukui, “Characteristics of whispering gallery modes in single spheroid: Dependence on the direction of incident light,” J. Korean Phys. Soc. 47, S38–S42 (2005).

M. Fujii, M. Haraguchi, T. Okamoto, and M. Fukui, “Characteristics of whispering gallery modes in single dielectric spheroid excited by gaussian beam,” Jpn. J. Appl. Phys. 44, 4948 (2005).
[Crossref]

Haus, H. A.

Heebner, J. E.

Hell, S. W.

K. Kolmakov, V. N. Belov, J. Bierwagen, C. Ringemann, V. Muller, C. Eggeling, and S. W. Hell, “Red-emitting rhodamine dyes for fluorescence microscopy and nanoscopy,” Chem. Eur. J. 16, 158–166 (2010).
[Crossref]

Henderson, M. R.

Hill, S. C.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[Crossref] [PubMed]

Himmelhaus, M.

M. Himmelhaus, S. Krishnamoorthy, and A. François, “Optical sensors based on whispering gallery modes in fluorescent microbeads: response to specific interactions,” Sensors 10, 6257–6274 (2010).
[Crossref] [PubMed]

Holler, S.

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Lett. 13, 3347–3351 (2013).
[Crossref] [PubMed]

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett. 28, 272–274 (2003).
[Crossref] [PubMed]

Hossein-Zadeh, M.

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering by a Sphere (Wiley-VCH Verlag GmbH, 2007), pp. 82–129.

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[Crossref]

Ilchenko, V. S.

Jacques, F.

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[Crossref]

Johnson, B. R.

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[Crossref]

Keng, D.

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Lett. 13, 3347–3351 (2013).
[Crossref] [PubMed]

Kerker, M.

H. Chew, P. J. McNulty, and M. Kerker, “Model for raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[Crossref]

Khoshsima, M.

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett. 28, 272–274 (2003).
[Crossref] [PubMed]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[Crossref]

Kippenberg, T. J.

A. Schliesser and T. J. Kippenberg, “Chapter 5 - cavity optomechanics with whispering-gallery mode optical micro-resonators,” in “Advances In Atomic, Molecular, and Optical Physics,”, vol. 58 of Advances In Atomic, Molecular, and Optical Physics, E. A. Paul Berman and C. Lin, eds. (Academic Press, 2010), pp. 207–323.
[Crossref]

Knight, J. C.

Kolchenko, V.

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Lett. 13, 3347–3351 (2013).
[Crossref] [PubMed]

Kolmakov, K.

K. Kolmakov, V. N. Belov, J. Bierwagen, C. Ringemann, V. Muller, C. Eggeling, and S. W. Hell, “Red-emitting rhodamine dyes for fluorescence microscopy and nanoscopy,” Chem. Eur. J. 16, 158–166 (2010).
[Crossref]

Korystov, D.

D. L. McAuslan, D. Korystov, and J. J. Longdell, “Coherent spectroscopy of rare-earth-metal-ion-doped whispering-gallery-mode resonators,” Phys. Rev. A 83, 063847 (2011).
[Crossref]

Krishnamoorthy, S.

M. Himmelhaus, S. Krishnamoorthy, and A. François, “Optical sensors based on whispering gallery modes in fluorescent microbeads: response to specific interactions,” Sensors 10, 6257–6274 (2010).
[Crossref] [PubMed]

Ksendzov, A.

Kulkarni, R. P.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783–787 (2007).
[Crossref] [PubMed]

Kuo, P. S.

Lai, H. M.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[Crossref] [PubMed]

Laine, J.-P.

Lake, T.

A. Boleininger, T. Lake, S. Hami, and C. Vallance, “Whispering gallery modes in standard optical fibres for fibre profiling measurements and sensing of unlabelled chemical species,” Sensors 10, 1765–1781 (2010).
[Crossref] [PubMed]

Leung, P. T.

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[Crossref] [PubMed]

Liang, W.

Libchaber, A.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
[Crossref]

Lin, Y.

Little, B. E.

Longdell, J. J.

D. L. McAuslan, D. Korystov, and J. J. Longdell, “Coherent spectroscopy of rare-earth-metal-ion-doped whispering-gallery-mode resonators,” Phys. Rev. A 83, 063847 (2011).
[Crossref]

Mackowski, S.

Maleki, L.

Marki, C.

C. Marki and S. D. E. E. P. University of California, Design and Optimization of Bidirectional and Optical Logic Systems in the Presence of Noise (University of CaliforniaSan Diego, 2007).

Matsko, A. B.

McAuslan, D. L.

D. L. McAuslan, D. Korystov, and J. J. Longdell, “Coherent spectroscopy of rare-earth-metal-ion-doped whispering-gallery-mode resonators,” Phys. Rev. A 83, 063847 (2011).
[Crossref]

McNulty, P. J.

H. Chew, P. J. McNulty, and M. Kerker, “Model for raman and fluorescent scattering by molecules embedded in small particles,” Phys. Rev. A 13, 396–404 (1976).
[Crossref]

Min, B.

B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457, 455–458 (2009).
[Crossref] [PubMed]

Monifi, F.

F. Monifi, S. Odemir, J. Friedlein, and L. Yang, “Encapsulation of a fiber taper coupled microtoroid resonator in a polymer matrix,” IEEE Photon. Technol. Lett. 25, 1458–1461 (2013).
[Crossref]

Monro, T. M.

Muller, V.

K. Kolmakov, V. N. Belov, J. Bierwagen, C. Ringemann, V. Muller, C. Eggeling, and S. W. Hell, “Red-emitting rhodamine dyes for fluorescence microscopy and nanoscopy,” Chem. Eur. J. 16, 158–166 (2010).
[Crossref]

Naweed, A.

S. I. Shopova, G. Farca, A. Naweed, and A. T. Rosenberger, “Whispering-gallery-mode microlaser consisting of a HgTe-quantum-dot-coated microsphere,” in Proceedings of Frontiers in Optics (2003), p. TuT4.

Nitzan, A.

J. Gersten and A. Nitzan, “Spectroscopic properties of molecules interacting with small dielectric particles,” J. Chem. Phys. 75, 1139–1152 (1981).
[Crossref]

J. Gersten and A. Nitzan, “Electromagnetic theory of enhanced Raman scattering by molecules adsorbed on rough surfaces,” J. Chem. Phys. 73, 3023–3037 (1980).
[Crossref]

Odemir, S.

F. Monifi, S. Odemir, J. Friedlein, and L. Yang, “Encapsulation of a fiber taper coupled microtoroid resonator in a polymer matrix,” IEEE Photon. Technol. Lett. 25, 1458–1461 (2013).
[Crossref]

Okamoto, T.

M. Fujii, M. Haraguchi, T. Okamoto, and M. Fukui, “Characteristics of whispering gallery modes in single dielectric spheroid excited by gaussian beam,” Jpn. J. Appl. Phys. 44, 4948 (2005).
[Crossref]

M. Fujii, M. Haraguchi, T. Okamoto, and M. Fukui, “Characteristics of whispering gallery modes in single spheroid: Dependence on the direction of incident light,” J. Korean Phys. Soc. 47, S38–S42 (2005).

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[Crossref]

Ostby, E.

B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457, 455–458 (2009).
[Crossref] [PubMed]

Ou, X.

D. Xiao-Wei, L. Shao-Hua, F. Su-Chun, X. Ou, and J. Shui-Sheng, “All-fibre micro-ring resonator based on tapered microfibre,” Chin. Phys. B 17, 1029 (2008).
[Crossref]

Prock, A.

R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 65 (1978).

Quan, H.

H. Quan and Z. Guo, “Simulation of whispering-gallery-mode resonance shifts for optical miniature biosensors,” J. Quant. Spectrosc. Radiat. Trans. 93, 231–243 (2005).
[Crossref]

Ramsey, J. M.

M. D. Barnes, W. B. Whitten, S. Arnold, and J. M. Ramsey, “Homogeneous linewidths of rhodamine 6G at room temperature from cavity-enhanced spontaneous emission rates,” J. Chem. Phys. 97, 7842–7845 (1992).
[Crossref]

Ringemann, C.

K. Kolmakov, V. N. Belov, J. Bierwagen, C. Ringemann, V. Muller, C. Eggeling, and S. W. Hell, “Red-emitting rhodamine dyes for fluorescence microscopy and nanoscopy,” Chem. Eur. J. 16, 158–166 (2010).
[Crossref]

Rosenberger, A. T.

S. I. Shopova, G. Farca, A. Naweed, and A. T. Rosenberger, “Whispering-gallery-mode microlaser consisting of a HgTe-quantum-dot-coated microsphere,” in Proceedings of Frontiers in Optics (2003), p. TuT4.

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[Crossref]

Rowland, K. J.

A. François, K. J. Rowland, V. S. Afshar, M. R. Henderson, and T. M. Monro, “Enhancing the radiation efficiency of dye doped whispering gallery mode microresonators,” Opt. Express 21, 22566–22577 (2013).
[Crossref] [PubMed]

A. François, K. J. Rowland, and T. M. Monro, “Highly efficient excitation and detection of whispering gallery modes in a dye-doped microsphere using a microstructured optical fiber,” Appl. Phys. Lett. 99, 141111 (2011).
[Crossref]

Ruppin, R.

R. Ruppin, “Decay of an excited molecule near a small metal sphere,” J. Chem. Phys. 76, 1681–1684 (1982).
[Crossref]

S. Afshar, V

S. Afshar, V.

Sáenz, J. J.

Savchenkov, A. A.

Schliesser, A.

A. Schliesser and T. J. Kippenberg, “Chapter 5 - cavity optomechanics with whispering-gallery mode optical micro-resonators,” in “Advances In Atomic, Molecular, and Optical Physics,”, vol. 58 of Advances In Atomic, Molecular, and Optical Physics, E. A. Paul Berman and C. Lin, eds. (Academic Press, 2010), pp. 207–323.
[Crossref]

Schmidt, M. K.

Seidel, D.

Shao-Hua, L.

D. Xiao-Wei, L. Shao-Hua, F. Su-Chun, X. Ou, and J. Shui-Sheng, “All-fibre micro-ring resonator based on tapered microfibre,” Chin. Phys. B 17, 1029 (2008).
[Crossref]

Shi, H.

J. D. Suter, I. M. White, H. Zhu, H. Shi, C. W. Caldwell, and X. Fan, “Label-free quantitative DNA detection using the liquid core optical ring resonator,” Biosens. Bioelectron. 23, 1003–1009 (2008).
[Crossref]

Shopova, S. I.

S. I. Shopova, G. Farca, A. Naweed, and A. T. Rosenberger, “Whispering-gallery-mode microlaser consisting of a HgTe-quantum-dot-coated microsphere,” in Proceedings of Frontiers in Optics (2003), p. TuT4.

Shui-Sheng, J.

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[Crossref] [PubMed]

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D. Xiao-Wei, L. Shao-Hua, F. Su-Chun, X. Ou, and J. Shui-Sheng, “All-fibre micro-ring resonator based on tapered microfibre,” Chin. Phys. B 17, 1029 (2008).
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J. D. Suter, I. M. White, H. Zhu, H. Shi, C. W. Caldwell, and X. Fan, “Label-free quantitative DNA detection using the liquid core optical ring resonator,” Biosens. Bioelectron. 23, 1003–1009 (2008).
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T. Søndergaard and B. Tromborg, “General theory for spontaneous emission in active dielectric microstructures: example of a fiber amplifier,” Phys. Rev. A 64, 033812 (2001).
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B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457, 455–458 (2009).
[Crossref] [PubMed]

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B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457, 455–458 (2009).
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A. Boleininger, T. Lake, S. Hami, and C. Vallance, “Whispering gallery modes in standard optical fibres for fibre profiling measurements and sensing of unlabelled chemical species,” Sensors 10, 1765–1781 (2010).
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M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9, 933–939 (2014).
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J. D. Suter, I. M. White, H. Zhu, H. Shi, C. W. Caldwell, and X. Fan, “Label-free quantitative DNA detection using the liquid core optical ring resonator,” Biosens. Bioelectron. 23, 1003–1009 (2008).
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M. D. Barnes, W. B. Whitten, S. Arnold, and J. M. Ramsey, “Homogeneous linewidths of rhodamine 6G at room temperature from cavity-enhanced spontaneous emission rates,” J. Chem. Phys. 97, 7842–7845 (1992).
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D. Xiao-Wei, L. Shao-Hua, F. Su-Chun, X. Ou, and J. Shui-Sheng, “All-fibre micro-ring resonator based on tapered microfibre,” Chin. Phys. B 17, 1029 (2008).
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F. Monifi, S. Odemir, J. Friedlein, and L. Yang, “Encapsulation of a fiber taper coupled microtoroid resonator in a polymer matrix,” IEEE Photon. Technol. Lett. 25, 1458–1461 (2013).
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F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics 1(3–4) 267–291 (2012).
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B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457, 455–458 (2009).
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H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
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B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457, 455–458 (2009).
[Crossref] [PubMed]

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J. D. Suter, I. M. White, H. Zhu, H. Shi, C. W. Caldwell, and X. Fan, “Label-free quantitative DNA detection using the liquid core optical ring resonator,” Biosens. Bioelectron. 23, 1003–1009 (2008).
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R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 65 (1978).

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F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002).
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A. François, K. J. Rowland, and T. M. Monro, “Highly efficient excitation and detection of whispering gallery modes in a dye-doped microsphere using a microstructured optical fiber,” Appl. Phys. Lett. 99, 141111 (2011).
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J. D. Suter, I. M. White, H. Zhu, H. Shi, C. W. Caldwell, and X. Fan, “Label-free quantitative DNA detection using the liquid core optical ring resonator,” Biosens. Bioelectron. 23, 1003–1009 (2008).
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D. Xiao-Wei, L. Shao-Hua, F. Su-Chun, X. Ou, and J. Shui-Sheng, “All-fibre micro-ring resonator based on tapered microfibre,” Chin. Phys. B 17, 1029 (2008).
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F. Monifi, S. Odemir, J. Friedlein, and L. Yang, “Encapsulation of a fiber taper coupled microtoroid resonator in a polymer matrix,” IEEE Photon. Technol. Lett. 25, 1458–1461 (2013).
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M. Fujii, M. Haraguchi, T. Okamoto, and M. Fukui, “Characteristics of whispering gallery modes in single spheroid: Dependence on the direction of incident light,” J. Korean Phys. Soc. 47, S38–S42 (2005).

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Nano Lett. (1)

V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano Lett. 13, 3347–3351 (2013).
[Crossref] [PubMed]

Nanophotonics (1)

F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics 1(3–4) 267–291 (2012).
[Crossref]

Nat. Methods (1)

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: Label-free detection down to single molecules,” Nat. Methods 5, 591–596 (2008).
[Crossref] [PubMed]

Nat. Nanotechnol. (1)

M. D. Baaske, M. R. Foreman, and F. Vollmer, “Single-molecule nucleic acid interactions monitored on a label-free microcavity biosensor platform,” Nat. Nanotechnol. 9, 933–939 (2014).
[Crossref] [PubMed]

Nature (1)

B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface-plasmon-polariton whispering-gallery microcavity,” Nature 457, 455–458 (2009).
[Crossref] [PubMed]

Opt. Express (4)

Opt. Lett. (5)

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[Crossref]

T. Søndergaard and B. Tromborg, “General theory for spontaneous emission in active dielectric microstructures: example of a fiber amplifier,” Phys. Rev. A 64, 033812 (2001).
[Crossref]

H. M. Lai, P. T. Leung, K. Young, P. W. Barber, and S. C. Hill, “Time-independent perturbation for leaking electromagnetic modes in open systems with application to resonances in microdroplets,” Phys. Rev. A 41, 5187–5198 (1990).
[Crossref] [PubMed]

Science (1)

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317, 783–787 (2007).
[Crossref] [PubMed]

Sensors (2)

A. Boleininger, T. Lake, S. Hami, and C. Vallance, “Whispering gallery modes in standard optical fibres for fibre profiling measurements and sensing of unlabelled chemical species,” Sensors 10, 1765–1781 (2010).
[Crossref] [PubMed]

M. Himmelhaus, S. Krishnamoorthy, and A. François, “Optical sensors based on whispering gallery modes in fluorescent microbeads: response to specific interactions,” Sensors 10, 6257–6274 (2010).
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C. Marki and S. D. E. E. P. University of California, Design and Optimization of Bidirectional and Optical Logic Systems in the Presence of Noise (University of CaliforniaSan Diego, 2007).

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[Crossref]

S. I. Shopova, G. Farca, A. Naweed, and A. T. Rosenberger, “Whispering-gallery-mode microlaser consisting of a HgTe-quantum-dot-coated microsphere,” in Proceedings of Frontiers in Optics (2003), p. TuT4.

K. Vahala, Optical Microcavities, Advanced series in applied physics (World Scientific, 2004).

A. Taflove, Computational Electrodynamics: The Finite - Difference Time - Domain Method, Antennas and Propagation Library (Artech House, Inc., 1995).

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Figures (7)

Fig. 1
Fig. 1 A circular flux collection region, which is offset from a 6 μm diameter microsphere in the x-direction, is placed so that its normal is aligned radially outward along the same axis. In this illustration, the optical modes are excited from a tangentially-oriented electric dipole source.
Fig. 2
Fig. 2 (a) A comparison of the power spectra of 6 μm diameter microspheres for different grid resolutions. (b) The convergence of the position of three central peaks is shown as a function of resolution. Excitation occurs from a tangentially oriented dipole source placed on the surface, with a central wavelength of λ = 0.6 μm. The flux collection time for each simulation is 0.6 ps.
Fig. 3
Fig. 3 (a) A comparison of the power spectra of 6 μm diameter microspheres with a tangential source (λ = 0.6 μm), for different flux collection times. (b) The behavior of the Q-factors of three central peaks is shown as a function of collection time (ps). The output power is normalized to the dipole emission rate in an infinite bulk medium of the same refractive index as the surrounding medium.
Fig. 4
Fig. 4 FDTD simulation of the normalized power spectrum of a polystyrene microsphere, 6 μm in diameter, with a surrounding medium of air. (a) Whispering gallery modes are excited from a tangential or (b) radial electric dipole source with a central wavelength of 0.6 μm. (c) The results for a surrounding medium of water are also shown for a tangential source and (d) a radial source. Vertical lines indicate predictions of the TEm,n (green) and TMm,n (red) modes derived from the Mie scattering analytic model, for azimuthal and radial mode numbers m and n, respectively. The width of the bands indicates the systematic uncertainty in the positions due to the finite grid size of FDTD.
Fig. 5
Fig. 5 A comparison of the Chew model of Eqs. (15) & (16) (dashed purple line) with the Mie scattering analytic model (vertical dashed lines) and the FDTD simulation, with total power collected from a box that surrounds the microsphere (solid blue line). A surrounding medium of water is used for a 6 μm diameter polystyrene sphere. (a) Whispering gallery modes are excited from a tangential or (b) a radial dipole source. The green vertical lines are the fundamental radial TE modes, and the red lines are the corresponding TM modes. The width of the bands accommodates the systematic uncertainty due to the finite grid size of FDTD.
Fig. 6
Fig. 6 Spatial distributions of the flux density over the collection region for several modes, S(r), integrated over a flux collection period of 1.0 ps, and projected onto the circular flux collection region. The modes considered are (a) TE40,1: 0.576 μm, (b) TE39,1: 0.588 μm, (c) TE38,1: 0.601 μm and (d) TE37,1: 0.615 μm. The axes are defined in the same way as in Fig. 1, for x-coordinate: 3.24 μm. Lighter color corresponds to larger magnitudes of S(r) · .
Fig. 7
Fig. 7 A comparison of the power spectra of 6 μm diameter microspheres with a tangential source (λ = 0.6 μm), (a) for flux collection regions at different distances Lflux from the centre of the sphere, and (b) for different flux region diameters Dflux. A resolution of Δx = 22 nm is used.

Tables (2)

Tables Icon

Table 1 Computing resources required for a three-dimensional FDTD simulation of a 6 μm diameter sphere excited by a dipole source with a central wavelength of 0.6 μm. The Tizard machine at eResearchSA (https://www.ersa.edu.au/tizard) is used in these simulations, which uses AMD 6238, 2.6 GHz CPUs. The number of CPUs, the memory (RAM), virtual memory (VM) and wall-time are listed for a variety of FDTD grid resolution values, Δx. The resolution in the frequency domain is quoted after being converted to wavelength, Δλ. The wall-time increases linearly with the flux collection time, which is held fixed at 0.6 ps in this table.

Tables Icon

Table 2 A summary of the Q-factors and wavelength positions, λ(μm), for the four most prominent WGM peaks, for each plot displayed in Fig. 4. The scenarios considered are: a surrounding medium of air or water with a tangential (Ey) or radial (Ex) dipole source. Due to finite collection time, a systematic uncertainty of up to 17% is expected in the Q-factors.

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

P ( λ ) = S n ^ d A ,
Q = λ 0 δ λ .
× × E 0 ( r ) k 2 n 2 ( r ) E 0 ( r ) = 0 ,
E 0 = S m ( r ) e i m φ k r X lm ( θ ) ,
X lm ( θ ) = i m sin θ P l m ( cos θ ) e ^ θ θ P l m ( cos θ ) e ^ φ ,
d S m ( r ) d r 2 + ( k 2 n 2 ( r ) m ( m + 1 ) r 2 ) S m ( r ) = 0 .
S m ( r ) = { z 1 r j m ( z 1 r ) , r < R A m z 2 r h m ( 1 ) ( z 2 r ) , r > R .
n 1 z 1 R ( m + 1 ) j m ( z 1 R ) z 1 R j m + 1 ( z 1 R ) j m ( z 1 R ) = n 2 z 2 R ( m + 1 ) y m ( z 2 R ) z 2 R y m + 1 ( z 2 R ) y m ( z 2 R ) .
E 0 = e i m φ k 2 n 2 ( r ) [ 1 r T m ( r ) r Y lm ( θ ) + 1 r 2 T m ( r ) Z lm ( θ ) ] ,
Y lm ( θ ) = e ^ r × X lm ( θ ) ,
Z lm ( θ ) = l ( l + 1 ) P l m ( cos θ ) e ^ r ,
d T m ( r ) d r 2 2 n ( r ) d n ( r ) d r d T m ( r ) d r + ( k 2 n 2 ( r ) m ( m + 1 ) r 2 ) T m ( r ) = 0 .
T m ( r ) = { z 1 r j m ( z 1 r ) , r < R B m z 2 r h m ( 1 ) ( z 2 r ) , r > R ,
1 n 1 z 1 R ( m + 1 ) j m ( z 1 R ) z 1 j m + 1 ( z 1 R ) j m ( z 1 R ) = 1 n 2 z 2 R ( m + 1 ) y m ( z 2 R ) z 2 y m + 1 ( z 2 R ) y m ( z 2 R ) .
P E y / P 0 E y = 3 ε 1 3 / 2 n 1 2 z 1 R 2 ( ε 2 μ 2 ) 1 / 2 m = 1 m ( m + 1 ) ( 2 m + 1 ) j m 2 ( z 1 R ) z 1 R 2 | D m | 2 ,
P E x / P 0 E x = 3 ε 1 3 / 2 n 1 4 z 1 R 2 ( ε 2 μ 2 ) 1 / 2 m = 1 ( 2 m + 1 ) [ | [ z 1 R j m ( z 1 R ) ] z 1 R D m | 2 + μ 1 μ 2 ε 1 ε 2 j m 2 ( z 1 R ) | D ˜ m | 2 ] ,
for D m = ε 1 j m ( z 1 R ) [ z 2 R h m ( 1 ) ( z 2 R ) ] ε 2 h m ( 1 ) ( z 2 R ) [ z 1 R j m ( z 1 R ) ] ,
D ˜ m = D m ( ε 1 , 2 μ 1 , 2 ) .

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