High-Q microresonators currently attract a great dealof attention because of their numerous applications intechnological and scientific fields such as the realiza-tion of microlasers, narrow filters, optical switching,ultrafine sensing, displacement measurements, high-resolution spectroscopy, and studies of nonlinear opti-cal effects. Several structures have been investigated,including whispering-gallery resonators,1–4 microringresonators,5–7 micropost (or micropillar) cavities,8 andphotonic crystal defect microcavities.9 Recently, a newstructure, the self-coupling nanowire coil resonatorwas proposed and investigated.10,11 The optical nano-wire microcoil resonator (ONMR) is a 3D generaliza-tion of the loop and ring resonators and can be createdby wrapping an optical nanowire on a low-index dielec-tric rod10,11; however, its geometry cannot be achievedby planar technology. Being fabricated from a single-mode fiber, the ONMR has a unique advantage incoupling light in and out of the resonator. With recentimprovements in fabrication technology of low-lossnanowires,12 the Q factor of ONMRs could potentiallycompete with the highest Q factors currently achievedonly in whispering-gallery resonators.13 Here we in-
vestigate the dependence of the Q factor of ONMRson the coupling parameter in three different typesof profile and demonstrate that the uniform (cylin-drical) profile is not optimal to easily obtain a highQ factor.
2. Coupled Wave Equations
Our analysis of the ONMR is based on the solution ofcoupled wave equations. Consider the propagation oflight along an ONMR, as illustrated in Fig. 1. Alongthe fiber, we define A��� as the light amplitude at theposition ��, R���� in cylindrical coordinates, whereR��� is the distance from the z axis, and � is the anglecoordinate. It is convenient to define the amplitude ofthe field at the mth turn as Am��� and to consider �as the common coordinate along turns, so that 0 �� � 2�. The propagation of light along the coil in anM turn ONMR is described by the coupled wave equa-tions for slowly varying R(�) (Ref. 10):
The authors are with the Optoelectronics Research Center,University of Southampton, Southampton SO17 1BJ, UK. F. Xu’se-mail address is [email protected].
Received 21 August 2006; accepted 15 September 2006; posted22 September 2006 (Doc. ID 74226); published 17 January 2007.
���Rq���d��, � is the propagation constant, and �pq isthe coupling coefficient11 between the pth and the qthturns. In the following, we consider only the couplingbetween adjacent turns, and suppose that the coeffi-cients � and � are independent of �. Field continuitybetween the turns implies the conditions
Rm1�0� � Rm�2��,
Am1�0� � Am�2��expi�0
2�
�Rm���ds,
m � 1, 2, . . . , M � 1.
We introduce the average coupling parameter K �2�R0�, where R0 � ��m�1
M �02� Rm���d���2�M is the
average radius, and the amplitude transmission co-efficient is defined as
T �AM�2��A1�0�
expi�0
2�
�RM���d� . (2)
3. Full Width at Half-Maximum and Q Factor inDifferent Profiles
If the propagation losses are ignored, � is real, and|T| � 1, i.e., the coil performs as an all-pass filter,and the resonances of the transmission coefficientappear only in the group delay. The Q factor nearwavelength 0 is defined as 0��, where � is theFWHM of the transmission spectrum or the groupdelay.14 In the lossless case, the Q factor becomesinfinite at resonant K, and the FWHM tends to zero.For the practical case of a nanowire with finite loss, onthe other hand, the FWHM has a nonzero minimum.The FWHM is related to K and to the ONMR profile.
We consider three simple fundamental profiles(Fig. 2) described by the following mathematicalformulas:
H �uniform�: Rm��� � R0,
V �conical�: Rm��� � R0 M2 dR �m � 1
�
2� dR,
X �biconical�: Rm��� � R0 ��M 12 � m �
� � �
2� ��dR �
M4 dR,
where m � 1, 2, . . . , M, dR�R0 �� 1, and |Rm1���� Rm���| � dR for any two adjacent turns.
For these profiles, we solve Eq. (1) numerically,assuming a nanowire diameter of D � 1000 nm, afiber refractive index of 1.46, an effective index ofneff � 1.2 at wavelength 1.55 m, R0 � 125�2 m,dR � 100 nm, propagation losses of 0.02 dB�mm,and K between zero and a maximum value KM. Kis very sensitive to the distance between the cen-tral axes of two adjacent turns: for our parameters,K � KM � 20 when the turns are touching (the dis-tance is D), and K � 4 for a distance of 1.5D.
Figure 3 shows the dependence of the FWHM on Knear 0 � 1.55 m for the three types of profile whenM � 3 in the whole range of K (0–20) on a linear scaleand near KM �14–20� on a logarithmic scale. We findthat the FWHM decreases monotonically with Kwhen K is very small �K � 1� and fluctuates widelywhen K � 1. Therefore, in general, higher Q factorscannot be obtained simply by increasing K, i.e., bymaking two adjacent turns closer. The FWHM of theH profile is periodic, and at K � KM is close to theminimum (approximately �0.02 nm). However, forother parameters, and in particular for M � 3, thetransmission spectrum and the FWHM fluctuate ape-riodically with K, and the FWHM can be very largeat K � KM (for example, we find a width of 0.5 nmfor M � 4). The minimum of the FWHM for M � 3of approximately 0.001–0.01 nm for all profiles isattributable to the finite loss of the nanowire. Inprinciple, the highest Q factor can be achieved byselecting a K for which the FWHM is minimized,but in practice this is difficult to realize because Kis too sensitive to the distance between adjacentturns. For ease of fabrication, it is therefore desir-able that the FWHM change slowly with K. Asshown in Fig. 3, an X-shaped resonator has a flatterFWHM than does an H, and H has a flatter one thandoes a V.
dd�
A1
A2
· · ·Am
· · ·AM�1
AM
� i
0R2����21���
0· · ·000
R1����12���0
R3����32���· · ·000
0R2����23���
0· · ·000
· · ·· · ·· · ·· · ·· · ·· · ·· · ·
000
· · ·0
RM�1����M�2M�1���0
000
· · ·RM�2����M�1M�2���
0RM����MM�1���
000
· · ·0
RM�1����M�1M���0
A1
A2
· · ·Am
· · ·AM�1
AM
,
(1)
Fig. 2. Illustration of three types of fundamental profiles ofONMRs: H (uniform), V (conical), and X (biconical).
To quantify the ease of obtaining high-Q-factor res-onators, we define the tolerance ratio as the fractionof K values where the FWHM is close to the minimumwithin a given interval. Specifically, we choose theintervals 0.9KM–KM, 0.8KM–KM, and 0.7KM–KM nearKM and look for FWHM � 0.01 nm. Note that K isrelated to the distance between adjacent turns, andthus changing K while keeping the geometry con-stant also implies changing dR. However, for the cho-sen ranges, this effect is small and has been neglectedin our calculations. Figure 4 shows the tolerance ra-tios for the specified ranges of K for ONMRs withM � 2–9, which are considered feasible targets withthe current technology. The tolerance ratios of H andX increase quickly with M: in particular, for X it isclose to 100% at M � 9. This suggests that it is easierto achieve high Q factors by fabricating as manyturns as possible. However, in practice it is much
more difficult to fabricate a resonator with manyturns to high accuracy. X is the optimal shape, and itstolerance ratio is nearly twice that of H for K between14 and 20. V is by far the worst geometry with atolerance ratio close to zero. Thus in order to fabricatehigh Q-factor ONMRs, the X profile provides a widerchoice of K to achieve optimal Q.
Finally, we investigate the tolerance ratio of the Xprofile as a function of dR when M � 3. Figure 5shows that the tolerance ratios increase quickly withdR, reach a maximum when dR � 0.05–0.2 m andthen decrease. Thus we find an optimum profile tofabricate X-shaped ONMRs. For the parametersconsidered here, this occurs at dR � 0.1 m. Exper-imentally, this can be achieved by etching a lowrefractive index rod on a very small slant (approx-imately 6°).
4. Summary
The coupled wave equations of ONMRs have beensolved for three profiles (uniform, conical, and biconi-cal). The tolerance ratio has been compared, and thebiconical geometry has proved to be optimal to fabri-cate high-Q resonators. Finally, for this profile, theoptimized slanting angle has been identified to be ofthe order of 6°.
The authors acknowledge the discussions with WeiLoh and David J. Richardson who initiated this re-search and the support provided for the work on thisproject by the Engineering and Physical SciencesResearch Council, under standard research grantEP�C00504X�1.
References1. B. Braginsky, M. L. Gorodetsky, and V. S. Ilchenko, “Quality-
factor and nonlinear properties of optical whispering-gallerymodes,” Phys. Lett. A 137, 393–397 (1989).
2. J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, “Phase-matched excitation of whispering-gallery mode resonances bya fiber taper,” Opt. Lett. 22, 1129–1131 (1997).
Fig. 3. FWHM dependency on the average coupling parameter Knear 0 � 1550 nm in three-turn ONMRs for three types of profile:H (dashed lines), V (dotted lines) and X (solid lines). (a) Linearscaling, K � 0–20; (b) logarithmic scaling, K � 14–20.
Fig. 4. Dependence of the tolerance ratio (defined as the percent-age of the average coupling parameter K where the FWHM isbelow 0.01 nm) near KM (14–20, 16–20, and 18–20) on the numberof turns M in a microcoil resonator for three types of profile: H(squares), V (triangles), and X (circles).
Fig. 5. Tolerance ratio near KM (K � 10–20, 14–20, and 18–20)versus dR in the x-profile three-turn ONMR.
3. V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki,“Dispersion compensation in whispering-gallery modes,” J.Opt. Soc. Am. A 20, 157–162 (2003).
4. S. M. Spillane, T. J. Kippenberg, O. J. Painter, and K. J.Vahala, “Ideality in a fiber-taper-coupled microresonator sys-tem for application to cavity quantum electrodynamics,” Phys.Rev. Lett. 91, 04902 (2003).
5. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J. P. Laine,“Microring resonator channel dropping filters,” J. LightwaveTechnol. 15, 998–1005 (1997).
6. D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, K. A.Stair, and S. T. Ho, “Waveguide-coupled AlGaAs�GaAs micro-cavity ring and disk resonators with high finesse and 21.6-nmfree spectral range,” Opt. Lett. 22, 1244–1246 (1997).
7. S. T. Chu, B. E. Little, W. Pan, T. Kaneko, S. Sato, and Y.Kokubun, “An eight-channel add-drop filter using verticallycoupled microring resonators over a cross grid,” IEEE Photon.Technol. Lett. 11, 691–693 (1999).
8. J. M. Gerard, D. Barrier, J. Y. Marzin, R. Kuszelewicz, L.
Manin, E. Costard, V. Thierry-Mieg, and T. Rivera, “Quantumboxes as active probes for photonic microstructures: The pillarmicrocavity case,” Appl. Phys. Lett. 69, 449–451 (1996).
9. R. K. Painter, A. Lee, A. Scherer, J. D. Yariv, P. O’Brien, D.Dapkus, and I. Kim, “Two-dimensional photonic bandgap de-fect mode laser,” Science 284, 1819–1821 (1999).
11. M. Sumetsky, “Uniform coil optical resonator and waveguide:transmission spectrum, eigenmodes, and dispersion relation,”Opt. Express 13, 4331–4340 (2005).
12. G. Brambilla, F. Xu, and X. Feng, “Fabrication of optical fibrenanowires and their optical and mechanical characterization,”Electron. Lett. 42, 517–519 (2006).
13. K. J. Vahala, “Optical microcavities,” Nature 424, 839–846(2003).
14. O. Schwelb, “Transmission, group delay, and dispersion insingle-ring optical resonators and add�drop filters—a tutorialoverview,” J. Lightwave Technol. 22, 1380–1394 (2004).
