1. science
2. physics
3. optics

Coherent interaction between two orthogonal
travelling-wave modes in a microdonut
resonator for filtering and buffering applications
Qingzhong Huang1,* and Jinzhong Yu1,2
1
Wuhan National Laboratory for Optoelectronics, Huazhong University of Science and Technology, Wuhan, 430074,
China
2
Institute of Semiconductors, Chinese Academy of Sciences, Beijing, 100083, China
*
[email protected]
Abstract: We theoretically investigate the coherent interaction between two
orthogonal travelling-wave modes in a microdonut resonator symmetrically
coupled to two bus waveguides. An analytical model has been developed to
describe this structure using transfer matrix method. The simulation reveals
that the two-mode microdonut can exhibit either a flat-top response or allpass transmission, governed by the resonance spacing. Then, we implement
analytical simulations to characterize the device and analyze the influence
of coupling efficiencies and propagation losses of two resonant modes on
behavior. Consequently, finite difference time domain simulations have
been performed. The numerical results validate our theoretical analysis, and
optical buffering effect is demonstrated in a pulse propagation simulation,
when the two resonances are aligned. In addition, we show that the device
function can be switched between flat-top filtering and all-pass filtering by
tuning the local refractive index in a microdonut. Hence, this structure is
promising for on-chip optical filtering and buffering applications.
©2014 Optical Society of America
OCIS codes: (130.3120) Integrated optics devices; (130.7408) Wavelength filtering devices;
(230.3990) Micro-optical devices; (230.5750) Resonators; (030.4070) Modes.
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(C) 2014 OSA
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1. Introduction
Coherent interaction between photonic resonators has attracted substantial interests in recent
years. Great efforts are made to map the coherent interference between excitation pathways in
atomic systems [1, 2] to resonances in optical resonator systems [3–16]. Optical analogues of
electromagnetic induced transparency (EIT) and Fano effect in atomic systems have been
studied based on coupled photonic crystal slabs [3, 4], planar photonic crystal cavities [5–9],
and whispering-gallery resonators [10–15]. As claimed in [16], coherent interference between
two resonant modes in a cavity generates intriguing characteristics. It can exhibit an EIT-like
or Fano-like response for two non-orthogonal modes, while behaving as either a flat-top filter
or an all-pass filter for two orthogonal modes. All-pass filters can provide lossless and strong
group delay, since they produce strong on-resonance phase variation while maintaining unity
transmission both on and off resonance. Unlike EIT-like and Fano-like effect, flat-top and allpass responses do not appear in an atomic system, indicating that the coherent interaction in a
multimode cavity is more versatile for practical applications.
By far, the reported two-mode cavities are constituted by one or two photonic crystal slabs
having two resonant modes of opposite symmetry for normally incident light [17, 18], or two
adjacent photonic crystal cavities side-coupled to a bus waveguide on a chip [19]. Here, we
employ a planar microdonut resonator [20, 21] coupled to two bus waveguides that can
support two orthogonal modes and has many merits with respect to filtering and buffering
applications. Firstly, the travelling-wave microdonut resonator behaves as an add-drop filter,
having the capability of dropping the forbidden wavelengths to another channel, instead of
reflecting them to the input port [22]. Secondly, this structure is suitable for on-chip photonic
integration, and the design is also flexible. Thirdly, the device function is switchable between
#222056 - $15.00 USD
Received 29 Aug 2014; revised 29 Sep 2014; accepted 29 Sep 2014; published 7 Oct 2014
(C) 2014 OSA
20 October 2014 | Vol. 22, No. 21 | DOI:10.1364/OE.22.025171 | OPTICS EXPRESS 25172
flat-top filtering and all-pass filtering via tuning the local refractive index in a microdonut
resonator by heating [15] or free-carrier injection/extraction [23].
In this paper, the coherent interaction of two orthogonal modes in a two-bus waveguides
coupled microdonut resonator is investigated analytically and numerically. An analytical
model is developed to describe this structure using transfer matrix method. It can function as
either a flat-top filter as the two resonances are spectrally detuned, or an all-pass filter as the
two resonances are aligned. Analytical simulations are implemented to characterize the device
and study the influences of coupling efficiencies and propagation losses of two whisperinggallery modes (WGMs) on performance. Consequently, we perform finite difference time
domain (FDTD) simulations, and the numerical results validate our theoretical analysis. As
the two resonances are aligned, optical buffering effect is demonstrated in a pulse propagation
simulation. Furthermore, it is shown that the function of device is switchable by tuning the
local refractive index in a microdonut resonator.
2. Structure, theory and principles
Figure 1(a) shows the schematic of a microdonut resonator symmetrically coupled to two bus
waveguides. The resonator can support two transverse modes, while the bus waveguide is
single mode. Here, we use the two lowest-order transverse modes in a microdonut resonator,
namely the first-order radial WGM (WGM1) and second-order radial WGM (WGM2).
WGM1/WGM2 and bus waveguide mode are evanescently coupled, while WGM1 and WGM2
are indirectly coupled in the bent-straight waveguide coupling region which can be regarded
as a directional 3 × 3 coupler. Hence, both WGM1 and WGM2 are possibly triggered by the
input optical field in the bus waveguide, and the two-mode interference will become
significant when their resonances are nearby. In fact, due to the perturbation in the coupling
region, the operating wavelength may shift slightly from the resonant wavelength [24]. For
simplicity, the influence of phase shifts in transmitted fields in the coupling region are not
considered in our model, as the field coupling is relatively weak.
Fig. 1. (a) Schematic of a two-mode microdonut resonator symmetrically coupled with two bus
waveguides. (b) General schematic of the interference between two coupling-out fields in the
drop channel as the resonances of WGM1 and WGM2 are aligned or appropriately detuned.
Using transfer matrix method, the input-output relations for a 3 × 3 coupler are given by
b0
t0
b1   jk1
b2
 jk 2
 jk1
 jk 2
a0
t1
kc
a1
 kc
t2
a2
(1)
where a0,1,2 and b0,1,2 are the optical fields (the subscripts 0, 1, and 2 represent the waveguide
mode, WGM1, and WGM2, respectively) at the input and output ports in the coupling-in
#222056 - $15.00 USD
Received 29 Aug 2014; revised 29 Sep 2014; accepted 29 Sep 2014; published 7 Oct 2014
(C) 2014 OSA
20 October 2014 | Vol. 22, No. 21 | DOI:10.1364/OE.22.025171 | OPTICS EXPRESS 25173
region, respectively; -jk1,2 and -kc are the field coupling coefficients between these modes;
t0,1,2 is the field transmission coefficient. The symbols c1,2 and d0,1,2 in Fig. 1(a) denote the
optical fields at the input and output ports in the coupling-out region, respectively.
It is seen that the output field in the drop channel is the interference result of the couplingout fields from two WGMs. The phase difference of the two coupling-out fields, associated
with the azimuthal order difference of two WGMs, plays an important role. The azimuthal
order of WGM1,2 is denoted by m1,2 = 2 Rneff1,2/ 1,2, where R, neff1,2 and 1,2 are the
microdonut outer radius, effective index and resonant wavelength of WGM1,2, respectively.
When (m1-m2) is even, the phase difference is 2p (p is an integer) and the interference is
constructive in the vicinity of the aligned resonances of non-orthogonal WGM1 and WGM2,
as studied in [25]. On the other hand, when (m1-m2) is odd, the phase difference is (2p + 1)
and the interference is destructive around the aligned two orthogonal modes. The conception
of “non-orthogonal” and “orthogonal” modes in a cavity coupled by multiple ports is
explained in [16]. Moreover, as the two resonances of WGM1 and WGM2 are spectrally
detuned, the phase difference changes, and the inter-mode interference may become
constructive for odd (m1-m2). In this paper, the subject of interest is the interference effect of
two WGM resonances and its applications as (m1-m2) is odd.
To describe the two-mode interference effect in our structure graphically, we consider an
operation wavelength at ( 1 + 2)/2 and assume two coupling-out fields of equal strength.
Figure 1(b) depicts the field coupling of WGMs and waveguide mode and the interference of
two coupling-out fields from the aligned or detuned two WGMs. When the two resonances
are rightly aligned, the two fields coupled out from on-resonant WGMs combine destructively
and cancel each other out, whereas when the resonances are detuned, the two fields coupled
out from off-resonant WGMs combine constructively and form a stronger field in waveguide.
The phenomenon suggests that this structure can possibly behave as either a flat-top filter or
an all-pass filter, which will be further explained in the following text.
We use = 2 2R(neff1 + neff2)/ to describe the phase shift of two-bus waveguides coupled
two-mode microdonut, where denotes the vacuum wavelength. The optical fields of WGMs
are phase shifted and cross-coupled in the resonator, and the relations are described as for (2n0.5) < < (2n + 0.5) , where n is an integer,
a1  b1 1 e j 1 t1  b2 ( 1 2 e j 1 e j 2 )1/ 2 ( kc )
a2  b2 2 e j 2 t2  b1 ( 1 2 e j 1 e j 2 )1/ 2 ( kc )
for (2n-1) <
(2n-0.5) or (2n + 0.5)
(2)
< (2n + 1) ,
a1  b1 1 e j 1 t1  b2 ( 1 2 e j 1 e j 2 )1/ 2 (kc )
a2  b2 2 e j 2 t2  b1 ( 1 2 e j 1 e j 2 )1/ 2 ( kc )
(3)
where 1,2 and 1,2 denote the round-trip field attenuation and phase shift for WGM1,2,
respectively, satisfying 1,2 = (4 2Rneff1,2)/ .
Then, the field transmissions in the through and drop channels are given by
St 
b0
a
a
 t0  jk1 1  jk 2 2
a0
a0
a0
(4)
d0
j 1 1/ 2 b1
j 2 1/ 2 b2
Sd 
  jk1 ( 1 e )
 jk 2 ( 2 e )
a0
a0
a0
We use Dr1 and Dr2 to denote the optical fields coupled out from WGM1 and WGM2,
corresponding to the first and second terms in the expression of Sd, respectively. Then, we are
able to obtain the wavelength-dependant time delay at the through port by
2
d
2 c d
(5)
#222056 - $15.00 USD
Received 29 Aug 2014; revised 29 Sep 2014; accepted 29 Sep 2014; published 7 Oct 2014
(C) 2014 OSA
20 October 2014 | Vol. 22, No. 21 | DOI:10.1364/OE.22.025171 | OPTICS EXPRESS 25174
where is the relative optical phase of St, and c is the light speed in vacuum.
To analyze the proposal more strictly, we simulate the output power spectra, as well as the
phase and time delay spectra at the through port by solving these equations. For simplicity, we
assume that WGM1 and WGM2 have an identical round-trip power attenuation ( 12 = 22) and
coupling efficiency to the waveguide mode (k12 = k22). In the simulation, we set k12 = k22 =
0.10, 12 = 22 = 0.999, 2 Rneff1 = 68.200 m and 2 Rneff2 = 59.675 m for WGM1 and
WGM2 in a microdonut resonator as an example, which are also generally consistent with the
parameters used in the FDTD simulations in Section 4. In this case, the two resonances of
WGM1 and WGM2 are rightly aligned at the wavelength of 1.705 m, and the corresponding
azimuthal orders are m1 = 40 and m2 = 35. The spacing between the two resonances can be
enlarged by simultaneously decreasing neff1 and increasing neff2 for detuned resonances.
Fig. 2. (a) Amplitude/power spectra of Dr1, Dr2, S t and Sd, and (b) phase spectra of Dr1, Dr2 and
St for the phase spacing
/ = 0.08. (c) Amplitude/power spectra of Dr1 , Dr2, S t and S d, and
(d) phase spectra of Dr1, Dr2 and St for
/ = 0. (e) Power spectra in the through and drop
channels, and (f) time delay spectra in the through channel for
/ ranging from 0 to 0.220.
As found in Fig. 2(a), flat-top responses are generated as the phase spacing
between
the peaks of Dr1 and Dr2 is 0.08 . Figure 2(b) shows that the phase difference between Dr1
and Dr2 varies around the two resonances, which is the minimum as the phase detuning /
is 0 and then increases to as | / | is enlarged. Hence, the output field at the drop port is
enhanced for the phase detuning between the two peaks due to the dominant constructive
interference, while it is reduced for the outside phase detuning due to the dominant destructive
interference. That is the reason the passband in the drop channel has a flat top and a steep rolloff. The stopband in the through channel also has a flat top and steep side walls, since the
output powers at the through and drop ports are complementary in principle. The all-pass
transmission in the through channel can be explained in the same way when the two WGM
resonances are aligned, as shown in Fig. 2(c). Figure 2(d) illustrates that the phase difference
between Dr1 and Dr2 almost remains
throughout the spectra, leading to a broadband
destructive interference. Therefore, interference cancellation occurs in the drop channel,
meanwhile the resonant valley almost disappears in the through channel and thereby the
structure behaves as an all-pass filter. The spectra of phase and time delay at the through port
are given in Fig. 2(d). By gradually enlarging the phase spacing
, we observe the variation
#222056 - $15.00 USD
Received 29 Aug 2014; revised 29 Sep 2014; accepted 29 Sep 2014; published 7 Oct 2014
(C) 2014 OSA
20 October 2014 | Vol. 22, No. 21 | DOI:10.1364/OE.22.025171 | OPTICS EXPRESS 25175
trend of power and time delay spectra in Figs. 2(e) and 2(f), indicating that this structure can
be switched between an all-pass filter and a flat-top filter by adjusting the phase spacing.
3. Analytical simulation and characterization
The transmission and performance of this device are simulated analytically. The main factors
are coupling efficiencies (k12 and k22) and round-trip power attenuations ( 12 and 22), as well
as phase spacing ( ) that illustrated previously. Coupling efficiency is determined by the
phase-matching condition and effective coupling length of WGM and waveguide mode. It is
possible to achieve the desired k12 and k22 by adjusting the width of bus waveguide, spacing
gap between the waveguide and microdonut, and even the coupling length when a pulley
coupling scheme is employed [26]. However, it is conventionally difficult to adjust the roundtrip power attenuation, which mainly originates from bend radiation and roughness-induced
scattering. Though high-order WGM suffers higher radiation loss, similar 12 and 22 are still
achievable in experiment, as reported in [27].
Fig. 3. The power spectra in the (a) drop and (b) through channels as a function of phase
detuning under various coupling efficiencies. (c) IL, out-of-band rejection (black line), and
ERthr (blue line) of the passband vs. k1 2 and k22. (d) BW1dB, BW3dB, and shape factor of the
passband vs. k12 and k22.
3.1 Flat-top response and filtering
Figures 3(a) and 3(b) give the power transmissions at the drop and through ports under
various coupling efficiencies, respectively, as the two resonances are offsetted by
= 0.08 .
Other parameters are set as the same as those used in Fig. 2, while k12 and k22 are variable.
The interested passband in the drop channel is characterized by insertion loss (IL), out-ofband rejection, 1 dB bandwidth (BW1dB), 3 dB bandwidth (BW3dB), and shape factor (defined
as BW1dB/BW10dB). Extinction ratio (ERthr) is used for the stopband in the through channel. As
labeled in Fig. 3(a), here we define the out-of-band rejection of device as the lower of the two
at / = 0.2 and 0.2. It is seen that, despite of k12 = k22 and 12 = 22, the power spectra are
slightly asymmetrical, due to the difference between neff1 and neff2. A large ripple appears in
the passband when k12 and k22 are very low, as shown in the power spectrum under k12 = k22 =
0.02 for example. For real applications, the ripple should be as low as possible, so all the
characteristic quantities are given under k12 = k22 0.08 in Figs. 3(c) and 3(d) to permit a ripple
lower than 1 dB. We find that, as k12 and k22 increase, IL firstly maintains relatively constant
#222056 - $15.00 USD
Received 29 Aug 2014; revised 29 Sep 2014; accepted 29 Sep 2014; published 7 Oct 2014
(C) 2014 OSA
20 October 2014 | Vol. 22, No. 21 | DOI:10.1364/OE.22.025171 | OPTICS EXPRESS 25176
and then rises rapidly, the out-of-band rejection keeps decreasing, BW1dB firstly decreases and
finally rises slowly, BW3dB keeps increasing, meanwhile the shape factor is always degraded,
suggesting that box-like response deteriorates with the increase of k12 and k22. That is because
the increase of k12 and k22 brings about a more gentle roll-off for each WGM resonance and a
stronger interference between two resonances, which lower the ripple and make the passband
more flat at first, and degrade the box-like response at last. On the other hand, in the through
channel, one can see that there also exist optimized coupling efficiencies to obtain a box-like
stopband, and ERthr increases to the maximum and then declines, as k12 and k22 increase.
Then, we study the influence of the inequality in k12 and k22 or 12 and 22 with respect to
the flat-top filtering. The power spectra at the two output ports are simulated under k12 = 0.10
and a varied k22, while maintaining the other parameters, as shown in Figs. 4(a) and 4(b). We
find that the power responses in two channels become more asymmetrical when the inequality
in k12 and k22 gets more severe. With the increase of k22, the out-of-band rejection at / =
0.2 gets lower and turns to be lower than that at
/ = 0.2. It is noteworthy that
asymmetrical filtering is also applicable in optical communications, as stated in [28]. Seen in
Fig. 4(c), the out-of-band rejection reaches the highest, meanwhile the power response
becomes the most symmetrical, when k22 is a little lower than k12, because of the equal
bandwidths of two WGM resonances. On the other side, we simulate the power spectra at the
two output ports under 12 = 0.999 and a varied 22, as shown in Figs. 4(d) and 4(e). Similarly,
the output power responses become more asymmetrical when the inequality in 12 and 22
becomes more serious. However, the bandwidth of passband changes notably, while the outof-band rejection remains generally. It is observed in Fig. 4(f) that, IL of the passband
declines, and BW1dB and BW3dB both increase, as 22 increases. That is because the strength of
WGM2 resonance is reduced, the left side of the original flat-passband is lowered down, and
the lineshape is degenerated to be triangular-like, as the propagation loss of WGM2 increases.
Fig. 4. The power spectra at the (a) drop and (b) through ports as a function of phase detuning
under various k 22, and (c) IL and out-of-band rejection of the passband vs. k22 , when keeping
k1 2 = 0.10. The power spectra at the (d) drop and (e) through ports as a function of phase
detuning under various 22, and (f) IL, BW1dB, and BW3dB of the passband vs. 22 , when
keeping 12 = 0.999.
3.2 All-pass response and buffering
Figures 5(a) and 5(b) show the power and time delay spectra in the through channel under
various coupling efficiencies, respectively, as the two WGM resonances are aligned ( = 0).
As seen, the on-resonance power loss is quite low, so the structure is still regarded as an allpass filter. Ideal all-pass transmission is hardly achievable, since WGMs are never lossless in
practice. It is found that, when k12 and k22 are low, the resonant valley (peak) in the power
(time delay) spectrum is sharp and deep (high), whereas when k12 and k22 get high, the
#222056 - $15.00 USD
Received 29 Aug 2014; revised 29 Sep 2014; accepted 29 Sep 2014; published 7 Oct 2014
(C) 2014 OSA
20 October 2014 | Vol. 22, No. 21 | DOI:10.1364/OE.22.025171 | OPTICS EXPRESS 25177
resonant valley (peak) in the power (time delay) spectrum becomes broad and shallow (low).
That is because the output bandwidth increases with the bandwidths of the fully-overlapped
two WGM resonances which both increase with k 12 and k22. Moreover, it is found that the
resonant power valley is splitted in two for much higher k12 and k22. To characterize the
performance of all-pass filter, we use IL and maximum time delay, effective bandwidth of
delay (BWdel) defined in [29], and delay-bandwidth-product (DBP) [30], as shown in Figs.
5(c) and 5(d). Note that BWdel and the bandwidth used in DBP of different units are inherently
the same. One can observe that, as k12 and k22 increase, the maximum time delay declines
near-exponentially, IL decreases in general and approaches zero, BWdel increases nearlinearly, while DBP decreases slightly around 1.0, revealing that DBP is basically a constant
and generally unaffected by k12 and k22. The small fluctuations in the curve of IL indicate that
the increase of k12 and k22 possibly causes a broader linewidth together with a higher onresonance power loss in some cases, when the two resonances are coherently interfering.
Fig. 5. The (a) power and (b) time delay spectra in the through channel as a function of phase
detuning under various coupling efficiencies. (c) IL and maximum time delay vs. k12 and k 22.
(d) BWdel and DBP vs. k12 and k22.
Then, we study the influence of the inequality in k12 and k22 or 12 and 22 with respect to
the all-pass filtering of our structure. The power and time delay spectra in the through channel
are shown in Figs. 6(a) and 6(b), respectively, using k12 = 0.10 and a varied k22. One can see
that the power loss around the resonance is still very low, and the time delay decreases with
k22. It is interesting that EIT-like response appears, when k22 is quite different from k12. Figure
6(c) shows that IL and the maximum time delay both decrease with k22 in general, though
some minor fluctuations exist for IL, similar to that in Fig. 5(c). On the other hand, we only
change 22 and calculate the output transmission, as shown in Figs. 6(d) and 6(e), while
maintaining the other parameters, such as 12 = 0.999, and k12 = k22 = 0.10. We observe that
the resonant valley becomes wider and deeper, as 22 is decreasing and even different from
2
1 , since the balance is broken in the destructive interference between two coupling-out
fields. It is also found that the maximum time delay changes slightly with 22. Figure 6(f)
shows that IL and the maximum time delay both decrease monotonously when 22 increases
from 0.790 to 0.999.
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Received 29 Aug 2014; revised 29 Sep 2014; accepted 29 Sep 2014; published 7 Oct 2014
(C) 2014 OSA
20 October 2014 | Vol. 22, No. 21 | DOI:10.1364/OE.22.025171 | OPTICS EXPRESS 25178
Fig. 6. The (a) power and (b) time delay spectra in the through channel as a function of phase
detuning under various k 22, and (c) IL and maximum time delay vs. k22, when keeping k 12 =
0.10. The (d) power and (e) time delay spectra as a function of phase detuning under various
2
2
2
2 , and (f) IL and maximum time delay vs. k 2 , when keeping 1 = 0.999.
4. FDTD validation and discussion
Consequently, 2-D FDTD simulations are performed to validate the analytical modeling and
scrutinize the two-mode interference in the proposed structure. Throughout the simulations,
slab waveguides are set with cladding and core refractive indices of 1.45 and 2.5,
respectively, which can be obtained for a silicon nanophotonic wire buried in silicon oxide
using effective index method. The gap between the microdisk and bus waveguide is 0.15 m,
R is 5 m, and the input light is transverse-magnetic polarized. Perfect match layer boundary
condition is used, and the grid sizes are x = y = 10 nm.
As shown in Fig. 7(a), WGM1 and WGM2 have been effectively excited, when the inner
radius (Ri) of microdonut and width of bus waveguide (W) are set as 3.50 m and 0.50 m,
respectively. We notice that the resonance of WGM3 is very weak and ignorable due to the
high bend radiation loss. When Ri increases to 3.55 m, we can observe in Fig. 7(b) that the
resonance of WGM2 is blue shifted, while the resonance of WGM1 almost holds the original
position, since WGM2 is more sensitive to the inner boundary of microdonut than WGM1, as
implied by the field distributions in Fig. 8(d). In the wavelength range of interest, the spacing
between two WGM resonances is the minimum near the wavelength of 1.60 m and increases
with the wavelength. It is seen that the extinction ratio of resonance in the through channel is
the smallest near 1.60 m, then it is enlarged to the maximum, after that the band top becomes
flat, and finally the flat-top stopband is splitted into two valleys, as the resonance spacing
increases with wavelength, consistent with the theoretical prediction in Fig. 2(e). Figure 7(c)
presents the power transmissions for Ri = 3.55 m and W = 0.47 m. The coupling efficiency
is influenced by waveguide width, since it depends on the mode matching between WGM and
waveguide mode. One can find that, all the resonances in Fig. 7(c) are barely shifted, while
the in-band ripples are reduced significantly, compared with those in Fig. 7(b), because of the
optimizing of coupling efficiencies. In addition, flat-top passband and stopband appear near
the wavelength of 1.67 m, due to the optimized resonance spacing, as seen in the inset of
Fig. 7(c). When Ri = 3.60 m and W = 0.50 m, the resonance of WGM2 is further shifted
leftward with respect to that in Fig. 7(b), and the two resonances of WGM1 and WGM2 are
nearly aligned at 1.705 m, bringing about all-pass filtering in the through channel, as shown
in Fig. 7(d). Note that the operation wavelength could be moved to a shorter wavelength such
as 1.550 m, by scaling down the device structure.
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Received 29 Aug 2014; revised 29 Sep 2014; accepted 29 Sep 2014; published 7 Oct 2014
(C) 2014 OSA
20 October 2014 | Vol. 22, No. 21 | DOI:10.1364/OE.22.025171 | OPTICS EXPRESS 25179
Fig. 7. The power transmissions simulated by FDTD methods for (a) Ri = 3.50 m, W = 0.50
m, (b) Ri = 3.55 m, W = 0.50 m, (c) Ri = 3.55 m, W = 0.47 m, and (d) Ri = 3.60 m, W
= 0.50 m. The blue and red lines represent the drop and through channels, respectively.
Fig. 8. The power transmissions for (a) Ri = 3.50 m, W = 0.50 m, (b) Ri = 3.55 m, W =
0.47 m, and (c) Ri = 3.60 m, W = 0.50 m. In (a), (b) and (c), the open circles and solid
black lines represent the FDTD and analytical simulations, respectively. (d) The field
distributions in two-bus coupled microdonut resonator corresponding to positions A, B, C, and
D in the transmission spectra.
Then, we implement analytical simulations to fit the FDTD data. As seen in Fig. 8(a), the
analytical fittings exhibit good agreement with the FDTD simulations, when we use k12 =
0.090, k22 = 0.070, 12 = 0.999, 22 = 0.996, neff1 = 2.17099, and neff2 = 1.90442. The modes
resonating at the wavelengths of 1.70517 m and 1.70931 m are identified as WGM(1,40) and
WGM(2,35), respectively, and the corresponding field distributions are provided in Fig. 8(d).
The value of k12, k22, 12, and 22 can be directly simulated by FDTD. The calculated k12 of
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Received 29 Aug 2014; revised 29 Sep 2014; accepted 29 Sep 2014; published 7 Oct 2014
(C) 2014 OSA
20 October 2014 | Vol. 22, No. 21 | DOI:10.1364/OE.22.025171 | OPTICS EXPRESS 25180
0.092, k22 of 0.073, 12 and 22 of higher than 0.990 are consistent with the derived values
from fitting. The analytical results fit the FDTD data well in Fig. 8(b), when k12 = 0.110, k22 =
0.090, 12 = 0.999, 22 = 0.996, neff1 = 2.17110, and neff2 = 1.90214. Moreover, the analytical
and FDTD results in Fig. 8(c) also keep consistent, when k12 = 0.100, k22 = 0.080, 12 = 0.999,
2
2 = 0.996, neff1 = 2.17064, and neff2 = 1.89946. The fitting results indicate that, as Ri is
enlarged, neff2 decreases greatly, while neff1 decreases slightly, agreeing with our prediction.
Flat-top and all-pass filtering have been realized, partially due to the similar k2 and 2 for two
WGMs. Furthermore, using the fitted parameters, a maximum time delay of 4.3 ps is
predicted for the all-pass filter [Fig. 8(c)].
The field distributions in Fig. 8(d) provide us more physical insight into the interference
effect and device behavior. As the two resonances are separate, the resonant mode is either
purely WGM1 or WGM2. Nevertheless, as the resonances are aligned, the resonant mode is a
hybridization of WGM1 and WGM2. Figure 8(d) shows that snake-like patterns of field are
produced in the microdonut when the two resonances are spectrally nearby or aligned, due to
the coherent interference between WGM1 and WGM2. The output power at the drop port
depends on the field distribution in coupling-out region. We find that the total coupling-out
field weakens seriously, when the resonances of WGM1 and WGM2 are aligned and a field
node appears in the coupling-out region. Meanwhile, the field in microdonut is the strongest,
as the field profile at position D shows, since the two WGMs both are on-resonance.
In the FDTD simulation, we launch a light pulse with a Guassian envelope and a carrier
wavelength of 1.705 m into the structure used in Fig. 8(c), and detect the output pulse, in
order to examine the pulse delay and signal waveform through the all-pass filter. Figures 9(a)
and 9(b) illustrate the light pulses passed through the straight waveguide with and without a
microdonut coupled, respectively. It is seen that the pulse has been delayed by (T2-T1) = 3.3
ps, approximately in accord with our analytical data based on the fitting parameters.
Furthermore, the output pulse power is attenuated by less than 10%, and no marked waveform
distortion is found, making it suitable for low-loss optical buffer application. The field
distributions in the structure at T1 and T2 are presented in the insets of Fig. 9. We observe the
optical field in straight waveguide being trapped into the microdonut at T1, and the field in
microdonut being released to the bus waveguide at T2, confirming the buffering effect of this
all-pass filter.
Fig. 9. (a) Temporal pulse shape detected by the monitor positioned at the output port of a
straight waveguide without a microdonut coupled. (b) Temporal pulse shape detected by the
monitor positioned at the output port of a straight waveguide coupled with a microdonut. T
denotes the time, and c is the light speed in vacuum. Insets of (a) and (b): the field distributions
in the microdonut resonator corresponding to cT1 and cT2 .
It is shown before that the resonance spacing is changeable by varying Ri. Here, by tuning
the refractive index in the inside lane of a microdonut, as depicted in the inset of Fig. 10(b),
we shift the WGM2 resonance significantly, while holding the WGM1 resonance. The change
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Received 29 Aug 2014; revised 29 Sep 2014; accepted 29 Sep 2014; published 7 Oct 2014
(C) 2014 OSA
20 October 2014 | Vol. 22, No. 21 | DOI:10.1364/OE.22.025171 | OPTICS EXPRESS 25181
of refractive index ( n) in the targeted area is achievable using local heating or free-carrier
injection/extraction. Based on the structure used in Fig. 8(c), we change the refractive index in
the inside lane, and calculate the power spectra using FDTD method, as seen in Figs. 10(a)
and 10(b). Figure 10(a) shows that the extinction ratio of combined resonance is improved,
and the all-pass transmission ( n = 0) is gradually transformed into a flat-top stopband
transmission ( n = 0.012), and then the single stopband is splitted into two, as n decreases
from 0 to 0.02. It validates that the proposed structure can be switched between a flat-top
filter and an all-pass filter, by dynamically tuning the local refractive index. Furthermore, the
power spectra confirm that the resonance of WGM2 can be shifted more substantially than
that of WGM2 and the resonance spacing is tunable using our scheme. Hence, it is expected
that this structure will find applications in flat-top passband/stopband filtering and low-loss
buffering.
Fig. 10. The power spectra in the (a) through and (b) drop channels under various n. Inset: the
schematic of a microdonut with a local refractive index change, where Ri = 3.60 m, Rd = 4.00
m.
5. Conclusion
In this paper, we have theoretically investigated the coherent interaction of two orthogonal
modes in a microdonut resonator coupled with two bus waveguides. An analytical model is
presented and employed to characterize this structure. It is found that when the two WGM
resonances are appropriately detuned, this device functions as a flat-top filter, whereas when
the two resonances are aligned, it functions as an all-pass transmission filter. Then, we have
studied the influence of coupling efficiencies and propagation losses of two WGMs with
respect to filtering and buffering performance. FDTD simulations are carried out to validate
the modeling and examine the two-mode interference in our structure. It is illustrated that the
numerical results exhibit good agreement with the theoretical fittings. As two resonances
aligned, optical buffering effect has been observed by letting a light pulse pass through the
structure in a FDTD simulation, and the time delay agrees with the theoretical prediction.
Furthermore, numerical simulations reveal that the function of device is switchable by tuning
the local refractive index in a microdonut resonator. Owing to the unique characteristics, the
proposed structure is expected to have wide applications in optical filtering and buffering.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grants
61006045 and 61177049, by the Major State Research Program of China under Grant
2013CB933303, and by the Major State Basic Research Development Program of China
under Grants 2013CB632104 and 2010CB923204.
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Received 29 Aug 2014; revised 29 Sep 2014; accepted 29 Sep 2014; published 7 Oct 2014
(C) 2014 OSA
20 October 2014 | Vol. 22, No. 21 | DOI:10.1364/OE.22.025171 | OPTICS EXPRESS 25182