Label-free, single-object sensing with a
microring resonator: FDTD simulation
Dan T. Nguyen* and Robert A. Norwood
College of Optical Sciences, University of Arizona, 1630 E. University Boulevard, Tucson, Arizona 85721, USA
*[email protected]
Abstract: Label-free, single-object sensing with a microring resonator is
investigated numerically using the finite difference time-domain (FDTD)
method. A pulse with ultra-wide bandwidth that spans over several resonant
modes of the ring and of the sensing object is used for simulation, enabling
a single-shot simulation of the microring sensing. The FDTD simulation not
only can describe the circulation of the light in a whispering-gallery-mode
(WGM) microring and multiple interactions between the light and the
sensing object, but also other important factors of the sensing system, such
as scattering and radiation losses. The FDTD results show that the
simulation can yield a resonant shift of the WGM cavity modes.
Furthermore, it can also extract eigenmodes of the sensing object, and
therefore information from deep inside the object. The simulation method is
not only suitable for a single object (single molecule, nano-, micro-scale
particle) but can be extended to the problem of multiple objects as well.
©2013 Optical Society of America
OCIS codes: (280.4788) Optical sensing and sensors; (280.1415) Biological sensing and
sensors; (140.3948) Microcavities; (080.1753) Computational methods.
References and links
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14 January 2013 / Vol. 21, No. 1 / OPTICS EXPRESS 49
1. Introduction
Recently, there has been great progress in the field of biosensors with tremendous sensitivity
down to single molecules or single particles of micron- or even nanometer scales [1–6]. The
enhancement mechanism responsible for this high sensitivity is the use of whispering gallery
modes (WGM) in optical micro-cavities such as microrings, microspheres and microdisks, in
which WGM resonance results in multiple interactions between the guided, re-circulating
light and the sensing object (SO). In such miniature sensors, which usually include a microcavity coupled to a waveguide (tapered fiber, for example), a single-shot measurement to
detect the SO is feasible. A light beam is launched from a remote location, propagates in the
waveguide, and then couples to the micro-cavity and thereby to a SO (single molecule or
particle), adjacent to the cavity. Due to WGM resonances, the light undergoes recirculation in
the cavity, and as a result, the light field interacts with the sensing object multiple times,
which can yield detailed information of the SO in the output transmission. The combination of
low-loss confinement of the light in the WGM cavity and recirculation of the light can
provide sensitivity down to single molecule or small particles of micron- and even
nanometers-scales. In 2002, Vollmer et. al. experimentally demonstrated enhancement of the
sensitivity using a silica microsphere with a radius ~100 micron [2]. However, even with this
enhanced sensitivity the resulting shift in the resonant wavelength induced by a single
molecule was very difficult to detect. Since then, progress has been made both experimentally
and theoretically to reach a sensitivity level that is high enough to enable detection of a single
molecule or a nanoparticle with R ~30nm [1–4]. In 2008, using a silica micro-toroid instead
of a microsphere, scientists achieved extremely high sensitivity resulting in detection of an
individual IL-2 protein molecule [3]. In 2010, nanoparticles with sizes down to 30nm have
been detected and analyzed using the mode splitting effect [4]. So far, most theoretical
analysis and calculations of WGM microrings and microsensors have been based on coupled
mode theory (CMT), which is straightforward and fast, based on approximations for simple
ideal structures which may not directly apply for real systems [1–7]. Usually, a simplified
theoretical analysis gives the shift of the WGM wavelength, assuming that there is a change
of the optical path in the cavity when it couples with a sensing object, where the change of the
optical path depends on the refractive index and the size of the sensing object. However, as
pointed out in Ref. [4], the shift is very small and susceptible to noise: intensity and frequency
noise of laser, thermal noise, detector noise and environmental disturbance. The shift also
depends on the strength of the coupling between the object and the microring WGM. As a
result a small particle with large coupling to the WGM can result in the same shift as a larger
object with smaller overlap. Furthermore, these analytical approaches fail to encompass many
significant and complex factors, such as the actual shape and eigenmodes of the SO,
determining the radiation losses of the light field in the tapered fiber and in the microcavity,
as well as the scattering loss in the system. All of these factors can be addressed through
FDTD simulation as presented in this paper. It is worth stressing that it is very difficult to use
CMT to describe the effect of multiple interactions between the light field with the SO. More
importantly, the FDTD can yield the resonant modes of the sensing object, which are easier to
observe than the small resonance wavelength shift predicted by CMT. The observation of the
eigenmodes of the SO has several important implications: (i) it is easy to detect the
eigenmodes of the SO by observing the transmission; (ii) once the eigenmodes are detected, it
is easy to determine the size and index of the SO, and (iii) the SO eigenmodes wavelengths do
not depend on the coupling strength between the WGM resonator and the SO which is an
important parameter in CMT, but is difficult to determine and control in experiment, and (iv)
the eigenmodes are almost immune to experimental noise sources, since it is the location of
the modes that is the most important, not their strength, which can be easily perturbed.
In this paper we present a finite-difference time domain (FDTD) simulation of microringbased sensing. Single-shot sensing of a label-free, single-object sensor using a microring is
simulated. A broad bandwidth pulse first propagates in a tapered fiber, couples to a resonant
microring and then to a single SO adjacent to the ring. Although the FDTD method can deal
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14 January 2013 / Vol. 21, No. 1 / OPTICS EXPRESS 50
effectively with different shaped objects with complex refractive indices that are applicable
for biosensing, for the sake of simplicity we consider in this paper only the basic cases in
which SOs are 2D particles (or microdisks) with radii rSO and real index nSO. The indices of
the SOs are chosen in the simulation as general examples to demonstrate the eigenmodes of
the objects with sizes of 1-2 micron within the frequency spectrum of 1.55 micron light
source. However, we would like to stress further that particles at the nano- and micro-scales
have increasingly become important objects in biosensing. The 2D SOs still represent some
aspects of the real particles in terms of the sensing properties, such as optical pathlength, and
the eigenmodes of the SO in the 2D plane. The results of FDTD simulation show the resonant
propagation in WGM microring, the multiple interactions between the recirculating light and
the SO, and the resonant light inside the SO. The microring transmission without a sensing
object (reference) is compared to that with sensing objects having different size and refractive
indices. As presented in the next section, our FDTD code allows us to simulate the microring
sensor with a high degree of flexibility, from designing a waveguide with minimum
reflection, to extracting information from inside microring and the sensing object as well. The
simulation method is not only suitable for a single object (single molecule, nano-, micro-scale
particle), but can also be extended to the problem of multiple objects as well. Our paper is
organized as follows: a general description of the system and the simulation method is
presented in Section 2, simulation results including animations of the sensing will be
presented in Section 3, and finally discussions of the advantages and disadvantages of the
FDTD method for modeling WGM sensing will be presented in Section 4.
2. General Description
Let us consider a typical microring sensor in which a waveguide (tapered fiber) couples to a
resonant microring adjacent to a sensing object (SO) as shown in Fig. 1. A light beam is
launched into the fiber from the left. The light beam propagates in the waveguide and then
couples to the microring, where WGM resonances enable multiple recirculation of the light,
resulting in multiple interactions between the light and the SO.
R
g
dR
dW
Fig. 1. Schematic of a microring sensor: a ring resonator (radius R, width dR and index nR)
coupled to a waveguide (width dW, index nW). The gap between the ring and the waveguide is
g. A sensing object (radius rSO and index nSO) is adjacent to the ring (light blue microdisk).
Note that, once the light beam is coupled to the microring its components having
frequencies that are close to WGM resonances of the cavity will be re-circulated multiple
times depending on the quality of the cavity Q. For example, in a planar microcavity with a
quality Q factor of 108, the resonant light can interact with the sensing object about 100,000
times, instead of only one interaction possible in a simple optical waveguide sensor [1–3]. By
monitoring the WGM optical resonances excited in the microcavity and/or the eigenmodes of
the sensing object, label-free, single-object sensing can be achieved with a single laser shot as
described below in our FDTD simulation.
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In this paper we investigate numerically the general problem of microring sensing for a
label-free single object sensing (SO) using the finite-difference time-domain (FDTD) method.
We simulate wideband pulse propagation in a waveguide coupled to a microring adjacent to a
sensing object. The FDTD method has several features that are advantageous for simulation
of such systems as described above. Furthermore, the FDTD method can completely describe
light recirculation in the WGM cavity and the multiple interactions between the light and the
SO. This feature is very unique to WGM sensing, and is almost impossible to describe
accurately by other simplified modeling methods. As presented below, FDTD simulation can
also yield the shifts of the resonant modes that are estimated through other simplified
modeling analysis [1–4]. More importantly, the FDTD simulation can extract the resonant
modes of the sensing object, which are in fact easier to detect than the small wavelength shift
of the microring modes. It is worth noting that FDTD method has been extensively applied to
simulate and analyze WGM of isolated microdisks, microdisks and microrings [8–10]. The
accuracy of the FDTD method for problems in linear optics was first demonstrated for the
directional coupler [11]. Since then, the perfectly matched layer (PML) absorbing boundary
condition (ABC) were introduced, providing the means to terminate the calculated grid space
with extremely low reflection [12].
Let us generally describe the FDTD simulation method for a typical label-free single-SO
microring sensor. We consider a two-dimensional (2D) problem where the z-directed electric
field is normal to the x-y plane of the grid. We employ the PML ABC in our simulation of the
light propagation in the waveguide-microring-SO system. It is worthwhile to stress here that
accurate FDTD simulations in resonant cavities in general, and especially in WGM cavities
have several challenges: (i) The computation time required for accurate simulation of light
recirculation in WGM cavities is much longer than that required for typical scattering or
propagation problems, and (ii) even if the reflection due to the numerical boundary conditions
is very small within PML ABC, its effects can adversely affect the simulation results. This is
especially the case when the light is reflected at the end of the waveguide, counter-propagates,
and then couples back to the microring, where it undergoes recirculation in the cavity. To
avoid boundary reflection during light recirculation in the microring, we introduce a long
waveguide around the sensing area, which includes the microring and the SO. The goal is to
make light keep propagating or trapped after it passes the microring so as to avoiding
reflection in the waveguide. The optimization challenge in this simulation is to design an
extended waveguide that can keep light propagating or get trapped as long as possible, while
at the same time minimizing the computational space needed.
(a)
-40
(b)
-20
0
20
40 μm
-40
-20
0
20
40 μm
Fig. 2. FDTD simulation of light field with frequency of 192 THz in a microring with R =
20μm, nR = nW = 1.46, dR = dW = 1μm . (a): without sensing object, and (b) with sensing object
rSO = 2 μm, nSO = 3.6. The long waveguide around the sensing system acts to minimize the
boundary reflection that can couple back into to the cavity during the resonant recirculation of
the light.
In this configuration, after traversing the coupling region between the waveguide and the
microring, the light keeps propagating in the extended waveguide, and at the end it trapped by
recirculation in the extended part of the waveguide.
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Figure 2 shows the intensity of the light with frequency f = 192 THz, which is resonant
with the microring, whose resonant modes are shown in Fig. 3. In the simulation, a fsec pulse
is launched from the left of waveguide and its ultrawide frequency band enables a single-shot
simulation that spans over several resonant modes of the microring and also some eigenmodes
of the sensing object. As shown in Fig. 2(b), in the case with a sensing object adjacent to the
ring, the light then couples to the object and excites its eigenmodes. The complex propagation
dynamics including multiple recirculation in the microring, and multiple interactions with the
sensing object will be simulated and shown in the animation discussed in next section.
Upon inspection of Fig. 2, we see that the effect of scattering due to the sensing object is
clearly shown (Fig. 2(b)) as compared to the microring without the object. More importantly,
the eigenmodes of the sensing object are clearly shown; the SO eigenmodes also appear in the
transmission spectrum and are distinct from those without a sensing object. The eigenmodes
are not only easier to detect compared with the shift of resonant modes in the measurement,
but furthermore are less influenced by noise and environmental disturbances which are
difficult to control or reduce in a micron-scale sensing system.
3. Numerical results
We consider in our simulation a microring with R = 20μm, dR = dW = 1μm and index nR = nW
= 1.46. Figure 3 shows typical results of a single-shot simulation.
(c)
(a)
0
2
0.5
1.0
1.5
2.0
2.5
3.0 3.5
Time (ps)
(d)
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Time (ps)
(b)
1
3
-20
0
20μ
188
190
192
194
Frequency (THz)
196
(e)
188
190
192
194
Frequency (THz)
196
(f)
0
0.5
1.0
1.5
2.0
Time (ps)
2.5
3.0 3.5
188
190
192
194
Frequency (THz)
196
Fig. 3. A general result from FDTD simulation of the pulse propagating in the waveguide and
coupling to the microring. (a) E-field of the input pulse in time domain (TD), and (b)
normalized intensity in frequency domain (FD). (c) E-field inside the cavity measured at
position 2 in TD, and (d) relative intensity in FD showing the resonant modes of the ring
without the sensing object. (e) E-field in TD and (f) transmission in FD (at position 3). Central:
intensity of the light with frequency f = 192 THz which is closest to a resonant mode of the
ring shown in (d).
The results shown in Fig. 3 represents a generalized example, in which the FDTD
simulation yields the input field at point #1, the field inside the cavity at point #2 and the
output field at point #3 in the time domain (TD) and frequency domain (FD). Later, in this
Section we will show the spectrum for each case under consideration, and discuss in details
the results. In the simulation, a femtosecond pulse with a cosine-modulated Gaussian
waveform is launched from the left into the waveguide as shown in Figs. 3(a) and 3(b) for the
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field E1(t) in the time domain (TD) and normalized intensity I1(f) = |E1(f)|2 in the frequency
domain (FD), respectively. The light field inside the microring (measured at position 2 of the
central figure) is shown in Figs. 3(c) and 3(d) for the field E2(t) in TD and relative intensity
I2(f) = |E2(f)|2/|E1(f)|2 in FD, respectively. The field measured at position 3 is shown in Figs.
3(e) and 3(f) for field E3(t) in TD and the transmission T(f) = |E3(f)|2/|E1(f)|2 in FD,
respectively. The central figure is the intensity of light with frequency f = 192 THz, which is
close to a resonant mode of the cavity as shown in Fig. 3(d) for the light intensity inside
cavity. Here Ej(t) is the electric field in the time domain at point j, and the Ej(f) are the Fourier
transform in frequency domain.
Now, let us investigate numerically how the light propagates in the sensing system with a
sensing object adjacent to the microring, and in particular determine what kind of information
the FDTD simulation can extract from sensing object. We performed FDTD simulations for
the system without SO, and with SOs having different sizes and indices as shown in Fig. 4 in
which the spectrum of a microring R = 20μm without an SO is compared with ones of the
same microring with adjacent SO r = 2, 3 μm (nSO = 3.6).
(a) w/o SO
Transmission T
1 .0
0 .8
0 .6
0 .4
0 .2
(b) SO r = 2μ
0 .0
w ith o u t S O
w ith S O r = 2 μ
w ith S O r = 3 μ
(d)
187 188 189 190 191 192 193 194 195 196 197
(c) SO r = 3μ
Relative Intensity I
2
F re q u e n c y (T H z )
5
w it h o u t S O
w it h S O r = 2 μ
w it h S O r = 3 μ
(e)
4
3
2
1
0
187 188 189 190 191 192 193 194 195 196 197
F re q u e n c y (T H z )
Fig. 4. Intensity of the light with f = 192 THz in the microring (R = 20, dR = 1 μm, n = 1.46)
with and without a sensing object. (a) without SO, (b) with SO r = 2 μm, nSO = 3.6, and (c)
with SO r = 3 μm, nSO = 3.6, (d) relative intensity inside cavity, (e) transmission: without SO
(red-curve), with SO r = 2 μm (blue), and with SO r = 3 μm (green); nSO = 3.6. Media 1 (or in
large format, Media 2) is a short animation of light propagation in the microring adjacent to SO
= 2micron described in Fig. 4(b).
Let us first discuss some main features of the results shown in Fig. 4. As shown in Fig.
4(a), light with frequency f = 192 THz is strongly resonant with the ring (R = 20μm, n = 1.46).
The frequency 192 THz actually is one of the resonant modes of the ring as shown by the red
curves in Figs. 4(d) and 4(e) for relative intensity inside the cavity and the transmission,
respectively, of the ring without SO. When a sensing object is included, a disk with diameter r
and index nSO, interesting effects occur including the following: (i) scattering by the SO, with
scattering intensity increasing with the size of the SO (see, Figs. 4(b), and 4(c)); the scattering
decreases intensity at f = 192 THz in the ring (Figs. 4(b) and 4(c)), as is also shown in Fig.
4(d) for the relative intensity inside the ring; (ii) Figs. 4(d) and 4(e) show the shifts of the
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resonant modes with an SO included, and the shift depends on the size SO (and, of course, the
refractive index); and (iii) it is clear that there are new modes that appear in the spectrum of
the system with the SO, and again the new modes depend on the physical properties of the
SO, such as size and refractive index.
To find out whether there is any relationship between the new modes of the sensing
spectra, e.g. Figures 4(d) and 4(e), we simulate the same light pulse coupling directly to the
SO and calculate the relative intensity inside the SO to see the eigenmodes of the SO within
the bandwidth of the light beam. Figure 5 shows the eigenmodes of several SOs with r = 1
and 2 μm with different indexes.
Fig. 5. Eigenmodes of several SOs with different size and indices within the bandwidth of the
light pulse that was used to simulate the microring sensing above. (a) r = 1μm, n = 3.6, (b) r =
2μm, n = 3.6, (c) r = 1μm, n = 4, and (d) r = 2μm, n = 2.0. The two SOs in (a) and (b) are used
for the simulation of microring sensing shown in Figs. 4(b) and 4(c) above. Media 3 (or in
large format, Media 4) is a short animation of light propagation in SO with r=2 m, n=3.6 (e.g.,
Fig. 5(b)).
The animations Media 1 and Media 3 below illustrate the time-domain simulation of the
sensing process in the entire system and the light in the SO, respectively. In Media 1, the
microring sensor was described in Fig. 4(c) and the resulting spectra are in Fig. 4(d) and 4(c).
The Media 3 shows the light coupling to the SO with rSO = 2μm and nSO = 3.6. Note that, due
to limit of space, we show here only short versions of the animations with only 1 or two
recirculation.
We have simulated different microring sensing systems with different ring sizes, and
sensing objects with different sizes and indices as some further examples shown in Fig. 6. The
results show that FDTD simulation yields very consistent results for the resonant shifts and
eigenmodes of the sensing objects. Figure 6(a) shows transmission T of a microring (R = 20
μm) without (red) and with SO r = 2μm (blue). It can be seen in Fig. 6 that there is a small
shift of the cavity resonant modes when the cavity is adjacent to an SO. In this case the cavity
light can couple to the SO. As a result, the optical pathlength increases and therefore shifts the
resonant modes in the cavity. More importantly, there are two new modes in the transmission
of the ring with an SO (indicated by two green arrows). Figure 6(b) shows the relative
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intensity I2 inside the microring without (red) and with SO (blue), and the eigenmodes of the
SO itself (green). Clearly, the two new modes in the transmission of the microring with SO
(blue curve in Fig. 6(a)) originated from the two eigenmodes of the SO.
Transmission T
1 .0
0 .8
0 .6
0 .4
0 .2
0 .0
187
w ith o u t S O
w ith S O r= 2 μ
188
189
190
191
192
193
194
195
(a)
196
197
Relative Intensity I
2
F re q u e n c y (T H z )
w ith o u t S O
w ith S O r = 2 μ
SO r = 2μ
5
(b)
4
3
2
1
0
187
188
189
190
191
192
193
194
195
196
197
F re q u e n c y (T H z )
Fig. 6. (a): Transmission T of the micro-sensing without (red) and without sensing object
(blue). The microring (R = 20 μm, dR = 1μm, nR = 1.46); SO rSO = 2 μm and nSO = 3.6., and (b):
Relative intensity inside microring without (red) and with (blue) sensing object, and inside the
sensing object (green) which show the resonant modes in the systems. The SO has rSO = 2 μm
and nSO = 3.6. The two green arrows indicate new modes that come from SO and can be
detected in the transmission.
Note that, the SO has two eigenmodes as shown in Fig. 6(b) (green curve): the first strong
mode is at ~192.8 THz and the second, but relatively weak mode is at ~189.9 THz. However,
the two corresponding modes in transmission spectrum (Fig. 6(a)) have quite similar
strengths. The main reason for this disproportion is that the second eigenmode at ~189.9 THz
has an overlap with a cavity mode at ~190.1 THz as shown in Fig. 6. Although the overlap is
weak, it enhances the apparent strength of the eigenmode in transmission as seen in Fig. 6(a).
3. Discussion
As discussed above, the FDTD method has some advantages for simulation of WGM sensing
systems. The FDTD simulation can describe many complex processes in the WGM sensing
system by using the real shape and eigenmodes of the SO, thereby enabling determination of
the radiation losses of the light field in the tapered fiber and in the micro-cavity, as well as the
scattering loss in the system. More importantly, FDTD can yield both the shift of the resonant
modes and the eigenmodes of the sensing object, which are easier to detect compared with the
small shift of the resonant modes. However, there are some disadvantages of the FDTD
method for simulation of WGM resonators in general and in particular for the problem of
WGM sensing. By far the most difficult challenge is managing the long computing time
required for simulation of the light recirculation in WGM resonators. It is widely accepted
that errors in FDTD simulations are negligibly small if the grid size of the computing space is
on the order of λ/10 to λ/20 [13]. However, this requirement is valid only for simulation of
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simple systems such as scattering problems in which there is no recirculation of resonant
light. In resonant cavities, the resonant frequency light undergoes recirculation multiple times,
and small numerical errors are accumulated during the long recirculation time, potentially
causing large errors in the final result. We have found that accurate simulations of the
microring require a grid size of λ/75 (λ = 1500nm, and grid size = 20nm). A combination of
small grid size and recirculation of resonant light in the WGM cavity requires large memory
and computing time. Boundary reflections make the situation even worse as longer
waveguides and larger computing space are necessary to avoid reflection. A simulation of a
typical microring sensing system with 16GB of RAM takes tens of hours.
(b)
Electric Field (a.u)
(a)
0
1
2
3
4
Time (ps)
5
6
7
0
1
2
3
Time (ps)
(c)
Fig. 7. Resonant modes of a microring with R = 20 μm and n = 1.46. (a) E field in time
domain, 10 recirculations, (b) E field in time domain, 5 recirculations, and (c) relative intensity
inside the microring in frequency domain: 5 recirculation (red) and 10 recirculation (blue).
Even though FDTD can accurately describe light recirculation in WGM microcavities,
accurate simulations of high Q-cavities in general and WGM sensing in particular are still
very challenging. The main reason for that is the huge number of times that the resonant light
recirculates in the WGM cavity. As mentioned in the introduction, an WGM microcavity with
Q of 108 has ~100,000 recicrulation cycles. Our simulations of typical microring sensors R =
20μm and about 10 recirculations in the cavity takes ~40 hours using an i7 PC with 16GB
RAM. Therefore, it is almost impossible to truly simulate the whole sensing process in a
microring sensing system since it would require unrealistic amounts of PC memory and is
also time consuming. It is important to stress that those limitations are difficult to overcome
for a single workstation. However, with the parallel computing using cluster PCs or
workstations, that problem can be significantly improved over what is presented in this paper.
Our FDTD simulations also confirm the physics of WGM resonances, i.e. that the higher the
number of light recirculation cycles, the narrower the width of the modes (the mode spacing is
not changed) and the stronger the resonances. In other words, the higher Q factor is
proportional to the number of recirculations or equivalently to the lifetime of the photon in the
cavity (or ~energy storage of the cavity). Figure 7 above shows the resonant modes in a
microring with R = 20μm and n = 1.46 for two simulations, one with 5 recirculation cycles
and the other with 10 recirculation cycles.
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It is important to note from the results in Fig. 7 that while the width of the modes changes
with the number of recirculations in the microring or essentially the computing time, the
central frequencies of the resonant modes do not change, and are clearly a fundamental
property of the microring. This important property allows us to use the resonant shifts from
the FDTD simulation to estimate the change in optical path due to the presence of the sensing
object. It is worthwhile to stress here that FDTD simulation can tackle a sensing object with
an arbitrary shape, a task much more difficult in other analyses. Furthermore, as presented
above, the eigenmodes of the sensing object from FDTD simulation provides additional
important information about the sensing object. It is also important to stress here that
“numerical” reflection from boundary can cause noisy spectra and even inaccurate results.
Therefore, the extended waveguide that traps the light after it traverses the microring is an
important feature. Good design can trap the light for a very long time, and at the same time
minimize the required computing space. Finally, we would like to stress that the conditions
for accurate simulation depend on the specific problems under consideration. For example, if
the SO size is smaller, the grid size should also be smaller so that the simulation would be
able to describe the eigenmodes of those SOs. Therefore, it is important to establish the
relationship between grid spacing and microring parameters, and the relationship between the
quality factor and the number of recirculations, a task that is the subject of ongoing work.
Finally, we would like to briefly discuss the results from Ref [4]. in which the authors
used the “mode splitting effect” to explain their experiments. In that paper, the new mode in
the spectrum of a microtoroid with a nanoparticle has been explained as the splitting of two
degenerate modes due to the appearance of the particle, and the scattering due to the
nanoparticle causes two standing waves in the cavity thereby lifting the degeneracy. First, it is
important to point out that the condition for degenerate modes in the microtoroid only hold for
an isolated microtoroid in which the system has perfect 2D symmetry. Under these conditions,
the appearance of the particle breaks the symmetry and lifts the degeneracy. In real
experiments, the microtoroid couples with a waveguide, the perfect 2D symmetry is broken
and therefore there are no longer any degenerate modes. Second, in the experiments, the
microtoroid was 5μm wide and the nanoparticle has a size of 30nm. For such a small particle,
scattered light escapes in many directions and an extremely small amount of light could be
coupled back to the microtoroid, therefore, its effect is very small even for the (perfect 2D
symmetry) of an isolated microtoroid. Even though standing waves can form in the
microtoroid due to scattering from one nanoparticle, the same is harder to realize if there are
many randomly placed particles in the microtoroid as described in [4].
Our FDTD simulation with much larger size SOs (micron-scale) cannot confirm that
standing waves occur due to scattering from the SO. It is simple to check in the FDTD
simulation whether there is counter-propagating light in the microring. During recirculation in
the microring, each time the light beam passes the coupling region (between the ring and the
waveguide), there is some light coupled to the waveguide with the rest continuing to circulate
as shown in Fig. 8.
-20
-10
0
10
20μ
-20
-10
0
10
20μ
-20
-10
0
10
20μ
Fig. 8. During the recirculation of light in the ring, each time it passes the coupling region,
light can couple to and subsequently co-propagates in the waveguide. Left: first time, (central)
second time, and (right) third time light passes the coupling region. (See the propagation of
light in the system for the whole time in the Media 1).
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Our simulation of the ring with more than 10 cycles did not evidence counter-propagating
light forming a standing wave due to scattering from the sensing object. We will further
investigate the WGM microring sensing system to see if the “splitting mode” or “eigenmode”
are dominant in actual experiments.
In conclusion, we have numerically investigated microring sensing using the FDTD
method. We considered the general problem of the FDTD simulation for WGM sensing, in
terms of a generalized sensing object. The results are for a single sensing object, but can be
extended to the case of many particles with different shapes. Our simulations show that the
FDTD method provides additional insight into the operation of WGM based sensors,
particularly with respect to the emergence of eigenmodes associated directly with sensing
objects, the shift of resonant microring modes, and other important information regarding the
eigenmodes of the sensing object.
Acknowledgments
This material is based upon work supported by the U.S. Air Force Office of Sponsored
Research under Award No. FA955010-1-0555 and upon the support of the CIAN NSF ERC
(EEC-0812072).
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Received 21 Sep 2012; revised 7 Dec 2012; accepted 11 Dec 2012; published 2 Jan 2013
14 January 2013 / Vol. 21, No. 1 / OPTICS EXPRESS 59