Coupling of whispering-gallery modes in size-mismatched microdisk
  photonic molecules
Microsoft Word - Boriskina_OL2007.DOC
Coupling of whispering-gallery modes in size-mismatched 
microdisk photonic molecules 
Svetlana V. Boriskina 
School of Radiophysics, V. Karazin Kharkov National University, Kharkov 61077, Ukraine 
Mechanisms of whispering-gallery (WG) modes coupling in microdisk photonic molecules (PMs) with slight and 
significant size mismatch are numerically investigated. The results reveal two different scenarios of modes interaction 
depending on the degree of this mismatch and offer new insight into how PM parameters can be tuned to control and 
modify WG-modes wavelengths and Q-factors. From a practical point of view, these findings offer a way to fabricate PM 
microlaser structures that exhibit low thresholds and directional emission, and at the same time are more tolerant to 
fabrication errors than previously explored coupled-cavity structures composed of identical microresonators. © 2007 
Optical Society of America. 
OCIS codes: 130.3120, 140.3410, 140.4780, 140.5960, 230.5750, 260.3160 
During the last decade, photonic molecules1 (clusters of 
electromagnetically-coupled optical microcavities) have gone 
a long way from a useful illustration of parallels between 
behavior of photons and electrons and now hold promise of 
new insights into physics of light-matter interactions and also 
of a variety of practical applications, including microlasers, 
tunable filters and switches, coupled-cavity waveguides, 
sensors, etc2-10. The simplest PM composed of two identical 
optical microcavities has been widely used as a test-bed to 
demonstrate shift and splitting of cavity modes and formation 
of a spectrum of bonding and anti-bonding PM supermodes1-
4. I have recently shown how arranging identical WG-mode 
microdisks into pre-designed high-symmetry configurations 
yields quasi-single-mode PMs with dramatically increased Q-
factors6, enhanced sensitivity to the environmental changes7, 
and/or directional emission patterns8. In all these structures, 
size uniformity of microcavities is an important issue in 
successful realization of PM-based optical components.  
The motivation for studying interactions of optical 
modes in a photonic molecule with size mismatch9,10 stems 
from two sources. First, precise and repeatable fabrication of 
identical microcavities, which in many cases are tiny 
structures having just several microns in diameter, is highly 
challenging. Second, a systematic study of double-cavity 
PMs with various degrees of cavities size mismatch can 
reveal new mechanisms of manipulating their optical 
properties thus paving way to improving or adding new 
functionalities to PM-based photonic devices. Such study has 
never been performed before, and is a focus of this letter. 
Despite its simplicity, the double-cavity structure provides 
useful insight into the general mechanisms of WG-modes 
coupling and offers new design ideas for more complex 
structures. The Muller boundary integral equations 
framework previously developed by the author to model PMs 
composed of identical cavities7 has been modified to study 
size-mismatched PMs. In the following, the term 
“microcavity mode” encompasses a complex value of the 
mode eigenfrequency and the corresponding eigenvector 
(i.e., modal spatial field distribution). 
The PM under study is composed of a pair of side-
coupled microdisks of radii RA, RB and refractive indices nA, 
nB separated by an airgap of width w (Fig. 1a). The 
microdisks of thicknesses much smaller than their 
diameters are considered. Thus, instead of the 3-D 
boundary value problem for the Maxwell equations, we 
solve its 2-D equivalent. In the following analysis, we 
search for the TE (transverse-electric) WG-modes, which 
are dominant in thin disks. At wavelength λ = 1.521 μm, a 
2-D cavity with radius 1.1 μm and effective refractive index 
63.2TE =effn  (2-D equivalent of a 200 nm-thick GaInAsP 
disk)2 supports WG8,1-mode with one radial field variation 
and eight azimuthal field variations (Q = 5243).  This mode 
(like all other WG-modes in circular cavities) is double-
degenerate due to the symmetry of the structure.  
WG-mode degeneracy is removed if two (or more) 
cavities are brought close together1-9, and four non-
degenerate WG-supermodes of different symmetry appear 
in the double-disk PM spectrum instead of every WG-mode 
of an isolated cavity. Fig. 2 (b and c) shows the wavelength 
migration and Q-factors change of these modes with the 
change of the radius of one of the cavities. The modes are 
labeled according to the symmetry of their field patterns 
along the y- and x- axes, respectively (e.g., OE supermode 
has odd symmetry with respect to y-axis and even 
symmetry with respect to x-axis). OE and OO modes are 
termed “anti-bonding” modes, while EO and EE modes are 
termed “bonding” ones. Bonding and anti-bonding 
supermodes group into nearly-degenerate doublets as seen 
in Fig. 1b. The values of real parts of eigenfrequencies of 
two modes in a doublet are so close to each other that they 
cannot be distinguished (Fig. 1b), while their imaginary 
parts differ, resulting in different Q-factors of these 
supermodes (Fig. 1c). Thus, in practice only two peaks are 
observed in a symmetrical double-cavity PM lasing 
spectrum (see Fig. 2 in Ref. 9), where the narrow high-
intensity peak corresponds to the high-Q anti-bonding mode 
doublet, and the wider low-intensity peak corresponds to the 
bonding one. 
0.8 1.0 1.2 1.4
(single disk)
Radius of microdisk B, R
 (μm)
 w (a) RA=1.1 μm RB 
Fig 1. (a) A geometry of a PM composed of two microdisks of different 
radii; (b) wavelengths migration and (c) Q-factors change of PM supermodes 
as a function of the radius of disk B (RA = 1.1 µm, w = 400 nm). The insets 
show the magnetic field distribution of the bonding (EE) and the anti-
bonding (OE) WG8,1 supermodes in the symmetrical (RA = RB = 1.1 µm) PM. 
Here and thereafter, corresponding characteristics of the WG8,1 mode of an 
isolated cavity with radius 1.1 µm are plotted for comparison (dash-dot line). 
1.05 1.10 1.15
1.05 1.10 1.15
 Superm
ode Q
-factor
Radius of microdisk B, R
 (μm)
Fig. 2. Supermodes wavelengths (left) and Q-factors (right) in the vicinity of 
anti-crossing point AC1 (RB = RA). Mode switching (see the modal near-field 
distributions at points A and B shown in the insets) at the anti-crossing point 
and Q-factor enhancement of one of the supermodes can be observed. 
Careful study of Figs. 1 b,c reveals a number of so-called 
exceptional points (corresponding to certain values of the 
varied parameter), where PM supermodes couple following 
either crossing (C) or avoided crossing (AC) scenarios. The 
behavior of wavelengths and Q-factors of the four 
supermodes in the vicinity of these exceptional points is 
shown in more detail in Figs. 2-4 (for the points AC1, AC2, 
and C1, respectively). The phenomenon of coupling of 
complex eigenvalues of matrices dependent on parameters 
under the change of these parameters is of a general nature 
and is observed in many physical systems11. Usually, 
frequency anti-crossing (crossing) is accompanied by 
crossing (anti-crossing) of the corresponding widths of the 
resonance states. Furthermore, at the points of avoided 
frequency crossing (points AC1-AC3 in Fig. 1), eigenmodes 
interchange their identities, i.e., Q-factors and field 
distributions.  
0.89 0.90 0.91 0.92
0.89 0.90 0.91 0.92
 Superm
ode Q
-factor
Radius of microdisk B, R
 (μm)
Fig. 3. Supermodes wavelengths (left) and Q-factors (right) in the vicinity 
of anti-crossing point AC2. Wavelengths repulsion accompanied by the 
linewidths crossing is observed. The insets demonstrate mode switching at 
the anti-crossing point (the modal near-field distributions shown in the 
upper(lower) insets correspond to the complex frequency values at the 
points labeled as A and D (B and C), respectively. 
In the context of coupling of WG-modes in microcavities, 
this interchange offers exciting new prospects for 
manipulating the PM optical characteristics, e.g. for 
realization of optical flip-flops9. For example, the coupling 
of modes with avoided frequency crossing scenario 
observed in Figs. 2 and 3 makes possible switching of field 
intensity between two microdisks. To realize such 
switching in practice, carrier-induced refractive index 
change of one of the disks induced by nonuniform pumping 
can be used. This effect was observed experimentally11 in a 
PM composed of nearly-identical microdisks (similar to the 
case shown in Fig. 2). If the microcavities are severely size-
mismatched, their WG-modes couple with the frequency 
crossing scenario. This situation is demonstrated in Fig. 4, 
and the numerical data indicate that such coupling may 
spoil significantly the Q-factors of the high-Q modes in the 
larger microdisk. However, optical coupling between 
cavities with strongly detuned WG-modes makes possible 
broad spectral transmission effects in coupled resonator 
optical waveguides (CROWs)10, coupled-resonator induced 
transparency12, and significant reducing of CROW bend 
radiation loses13. 
0.72 0.75 0.78
0.72 0.75 0.78
 Superm
ode Q
-factor
Radius of microdisk B, R
 (μm)
Fig. 4. Supermodes wavelengths (left) and Q-factors (right) in the vicinity of 
the crossing point C1. Wavelength crossing accompanied by damping of the 
high-Q supermodes is observed. The insets show supermodes near-field 
portraits at the crossing point. 
0.4 0.6 0.8
1.473
1.474
0.4 0.6 0.8
5x103
6x103
7x103
8x103
9x103
 Superm
ode Q
-factor
Disk-to-disk separation (μm)  
Fig. 5. Resonance wavelength (left) and Q-factor (right) of an anti-bonding 
WG-supermode in a three-disk PM. The central disk of radius 1.065 μm is 
separated from the side disks of radii 1.1 μm by airgaps of 400 nm width. 
Supermode near-field portrait and directional far-field emission pattern at the 
point labeled as A are shown in the inset. 
Finally, enhancement of the Q-factor of a single 
supermode in a double-disk PM in comparison to its single-
cavity value can be observed in Fig. 2 in a relatively wide 
range of cavities radii detuning: 14 nm < ΔR < 53 nm (ΔR = 
|RA - RB|). Note that all the other PM supermodes have 
significantly lower Q-factors in this range of the parameter 
change. This effect offers a way for selective enhancement of 
the Q-factor of a single supermode that (unlike symmetry-
enhanced Q-factor boost in polygonal PMs)6,7 does not rely 
on exact cavity size-matching. A possible realization of a 
PM-based structure designed by making use of this 
mechanism of selective mode enhancement is presented in 
Fig. 5. It consists of three coupled microcavities, with the 
central cavity radius detuned by ΔR = 35 nm from the side 
cavities radii. By adjusting the width of the airgaps between 
microcavities, noticeable Q-factor enhancement of one anti-
bonding supermode is achieved without shifting the 
supermode wavelength (Fig. 5). Furthermore, such PM 
demonstrates directional light emission, which cannot be 
achieved in isolated WG-mode microdisks (see inset to Fig. 
5). Our studies also indicate that this directional emission 
pattern is preserved if the disk-to-disk distance is varied. 
It should also be noted that other system parameters 
can be tuned to manipulate resonance wavelengths and Q-
factors of microcavities through mode coupling at 
exceptional points. Among these are: the refractive index of 
the cavity substrate, and the size and/or position of a hole 
pierced in the cavity, which can be adjusted to enhance a 
WG-mode Q- factor14,15 or to achieve directional emission 
on a high-Q WG-mode16. 
In summary, a comprehensive numerical study was 
performed to elucidate the mechanisms of modes coupling 
in PMs with various degrees of cavities size mismatch. The 
study offers an alternative approach to design novel PM-
based components with improved functionalities. In 
contrast to PM structures composed of identical cavities 
that may require fabrication accuracy beyond the 
capabilities of modern technology, the proposed approach 
does not rely on precise cavities size-matching to achieve 
the desired device performance. This approach paves the 
way for new designs of more complex PM structures and 
arrays, which may eventually lead to new capabilities and 
applications in micro- and nano-photonics. 
I wish to thank Vasily Astratov for discussions and Jan 
Wiersig for bringing his recent paper16 to my attention. This 
work has been partially supported by the NATO 
Collaborative Linkage Grant CBP.NUKR.CLG 982430. 
Svetlana Boriskina’s e-mail address is 
SBoriskina@gmail.com.  
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