Jimba, Y.

Kuwata-Gonokami, M.

Miyazaki, H.

Mukaiyama, T.

Takeda, K.

Physical Review Letters

English

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Tohoku University Repository (TOUR) / 東北大学機関リポジトリ

Tight-Binding Photonic Molecule Modes ofResonant Bispheres著者 宮嵜  博司journal orpublication titlePhysical review lettersvolume 82number 23page range 4623-4626year 1999URL http://hdl.handle.net/10097/35135doi: 10.1103/PhysRevLett.82.4623VOLUME 82, NUMBER 23 P HY S I CA L REV I EW LE T T ER S 7 JUNE 1999Tight-Binding Photonic Molecule Modes of Resonant BispheresT. Mukaiyama,1 K. Takeda,2 H. Miyazaki,3 Y. Jimba,4 and M. Kuwata-Gonokami1,*1Department of Applied Physics, the University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan,Cooperative excitation project ERATO, Japan Science and Technology Corporation (JST), Tokyo, Japan2Tsukuba Research Laboratory, JSR. Co., 25, Miyukigaoka, Tsukuba, Ibaraki, 305-0841, Japan3Department of Applied Physics, Tohoku University, Aramaki, Aoba08, 980-8579, Japan4College of Engineering, Nihon University, Koriyama, Fukushima, 963, Japan(Received 9 November 1998)We observe coherent resonant coupling of optical whispering-gallery modes in fluorescence from dyedoped polymer bispheres with diameters ranging from 2 to 5 mm. By monitoring the frequencies offluorescence peaks of individual spheres, we sort out two spheres with appropriate size matching andbring them into contact. Wave optics calculation also gives good agreement with the experiment. Bytaking into account harmonic coupling of the whispering-gallery modes, the obtained features of normalmode splitting are well explained by the tight-binding photon picture. [S0031-9007(99)09349-7]PACS numbers: 42.60.DaManipulation of light path in micrometer length scalehas recently attracted considerable attention from bothfundamental and application points of view. Convention-ally, the manipulation is based on the photonic crystalconcept [1–3]. In photonic crystals, which have peri-odic modulation of the refractive index, propagation ofthe light wave is governed by a weak potential. Cor-respondingly, such an approach can be referred to as anearly free photon approach analogous to the nearly freeelectron approach in band theory. Alternatively the mi-cromanipulation of light can be achieved by exploring thepossibility of confining the light in a small unit of thewavelength size. Light propagates through the system ofsuch units due to the coupling between the nearest neigh-bors. This approach is referred to as the tight-bindingphoton approach [4]. Within the tight-binding photon ap-proach we can guide the optical waves by connecting theunits in the arbitrarily shaped microstructures.The microspheres are the most natural choice of theunit to be employed in the tight-binding photon device.It is known that a dielectric sphere acts as a unique opti-cal microcavity which has very long photon storage timewithin a small mode volume [5–8]. In particular, Q fac-tors of the order of 1010 have been observed for whis-pering gallery modes (WGM’s) in quartz spheres with adiameter of several tens of micrometers [9–13], and themode structure of a pair of these large spheres in con-tact has been studied [14]. However, in order to explorethe feasibility of micromanipulation of light, one has toconfirm the existence of the coherent coupling betweenspheres of the size of a few times of optical wavelength.Lorenz-Mie theory predicts long photon lifetime even forsmall spheres, giving, for example, nearly 30 ps for a4 mm sphere with a refractive index of 1.59. This hasallowed one to propose such relatively small spheres to beemployed as “photonic atoms” [15] for the tight-bindingscheme. However, the coherent coupling between twoadjacent microspheres of such size range have not been0031-9007y99y82(23)y4623(4)$15.00realized until now. The coherent coupling results in thesplitting of the corresponding WGM’s and is a manifes-tation of the well-known phenomena of the normal modesplitting (NMS) in coupled harmonic oscillators. How-ever, although some attempts have been made [16], NMShas not yet been observed because of the difficulty in theprecise size control of the spheres.In this Letter, we report on the observation of normalmode splitting in the system of two polymer spheres incontact (bisphere) under extreme size control. We studybispheres formed from the spheres of diameters rangingfrom 2 to 5 mm. The sufficiently narrow linewidth andwide separation of WGM’s in this size range allowus to avoid intricate band mixing. The frequencies ofthe observed bisphere resonances agree with the waveoptics calculations. In order to examine the feasibility oftight-binding manipulation of light waves in a structurecomposed of connected microspheres, we estimate anintersphere coupling constant which is found to be largerthan the WGM linewidth.We use monodispersive polystyrene spheres (refrac-tive index 1.59) which are soaked with a solution ofdye (Nile Red, concentration is about 1022 molyl, fluo-rescence FWHM is about 70 nm). Dye doped spheresare sown on a glass plate under a microscope and can bemanipulated with an optical fiber probe which is made bypulling the multimode optical fiber in a frame. By using amicroscope objective lens, the individual sphere is excitedwith the second harmonics of a cw Q-switch Nd-YLFlaser sl ­ 527 nmd. The dye fluorescence is collectedwith the same fiber probe and sent to a spectrometer witha liquid nitrogen cooled CCD (charge-coupled device) de-tector. We use both 50 cm and 1 m focal length spec-trometers with spectral resolution of 0.06 and 0.03 nm,respectively.Figure 1 shows an example of fluorescence spectrumfrom the individual sphere of about 4.1mm in diameter.Sharp lines, which originate from the strong modification© 1999 The American Physical Society 4623VOLUME 82, NUMBER 23 P HY S I CA L REV I EW LE T T ER S 7 JUNE 1999FIG. 1. Typical fluorescence spectrum from 4.1 mm sphere.Calculated positions for s ­ 1 and s ­ 2 are shown at thebottom of the figure.of the vacuum field inside the sphere [6–8,17–19], appearat the resonance frequencies of WGM’s with TE and TMpolarizations.According to the Lorenz-Mie theory, each WGM can bespecified by the mode number n (indicating the numberof light wavelengths around the circumference), the ordernumber s (corresponding to the number of maxima in theradial dependence of the internal electric field), and theazimuthal mode number m (giving the orientation ofthe WGM’s orbital plane). Since the frequency of WGMdoes not depend on m, each resonance has s2n 1 1d-folddegeneracy.For spheres of diameter from 2 to 5 mm we can identifythe mode indices n and s of the high-Q WGM bycomparing the positions of the lines in the fluorescencespectra with the predictions of the Lorenz-Mie theory.The widths of the resonances in the fluorescence spectrumgive the actual Q factor of the particular sphere. Once welabel the fluorescence lines with the mode indices n ands, we can judge the relative size of the sphere within anerror of 1yQ which is about 0.05% for a 5 mm sphere.Spheres of 2 mm in diameter are found to have a Q factorof the order of 102. This is comparable with the radiativeleakage rate which follows from the Mie theory. Spheresof diameters more than 4 mm are found to have Q factorsof the order of 103. This is 2 orders less than the radiativeleakage rate [20].In order to make the bisphere, we choose two spheresof the desired size by comparing the frequencies of thespecific WGM resonances. By handling the optical fiberprobe attached to the three axis mechanical stage, we pickup one sphere and attach it to the other.We measure the bisphere fluorescence in parallel con-figuration [Fig. 2(a)-(A); the fiber probe is set parallelto the axis of the bisphere] and perpendicular configura-tion [Fig. 2(a)-(B); the fiber probe is set perpendicular tothe axis of the bisphere]. Figures 2(b) and 2(c) show thefluorescence spectra of individual spheres sCd and thebisphere sA, Bd in the vicinity of TE30,1 (TE mode ofn ­ 30 and s ­ 1) and TM29,1 (TM mode of n ­ 29and s ­ 1) resonances. For both the polarizations, we4624FIG. 2. (a) Microscope images of the bisphere and the fiberprobe. Diameter of the probe is about 5 to 10 mm. Wedetect the emission in two different geometries; one is parallelconfiguration (A), the other is perpendicular configuration (B).(b),(c) Spectra of resonance bisphere of TE30,1 mode andTM29,1 mode in parallel configuration sAd and perpendicularconfiguration sBd. Spectra sCd show the fluorescence ofindividual spheres before contact. Two spheres have almostthe same diameter. The arrows indicate the coupled modes.observe new peaks due to the intersphere coupling, in theparallel configuration sAd but not in the perpendicular con-figuration sBd. To explain this, one should recall that eachWGM of a given n and s has s2n 1 1d-fold degeneracyassociated with the azimuthal mode number m. Hence2n 1 1 orientations of the WGM orbital plane are permit-ted and, correspondingly, s2n 1 1d2 combinations of theintersphere couplings are generally possible. However,the intersphere coupling is maximum for the pair of modeswhose orbitals include the contact point and lay in thesame plane. Therefore, the fluorescence of the coupledmodes is maximum in the direction parallel to the bisphereaxis. As a result, the signal from the coupled interspheremodes is more pronounced in the parallel configurationthan in the perpendicular configuration. As we can seein Figs. 2(b) and 2(c), the uncoupled modes fluorescencealso contributes to the signal. By reducing the apertureof the fiber or keeping the probe away from the bisphere,this signal diminishes.The dependence of the mode coupling on the orbitalplane orientation breaks the degeneracy with respect tothe azimuthal indices. In particular, bisphere modesoriginating from the coupling of the WGM’s with thedifferent combinations of azimuthal numbers contributedifferently to the signal. Correspondingly, the line shapeof the signal represents the energy distribution among thecoupled modes. The asymmetric line shape of the coupledmodes on Figs. 2(b) and 2(c) indicate that the strongerthe coupling between WGM’s, the more the energy in theresulting bisphere mode.In order to study the dependence of the WGM couplingon the sphere size (size detuning), we fix the diameterof one sphere and change the diameter of the secondsphere in the scale of 0.3%. Figure 3 shows the detuningdependence of the bisphere fluorescence in the vicinity ofVOLUME 82, NUMBER 23 P HY S I CA L REV I EW LE T T ER S 7 JUNE 1999Fluorescence (arb.units)610609608607606605604603602Wavelength (nm)TE29,1FIG. 3. Emission spectra of nine bispheres with slightlydifferent sized spheres. Fiber probe is set parallel to the axis ofthe bisphere. Circles indicate the uncoupled modes. Trianglesshow the coupled modes of bisphere.TE29,1 mode for various pairs with size detuning. Circlesshow the uncoupled original resonance, and trianglesshow coupled modes. We clearly see the anticrossingbehavior which is the signature of NMS.Figure 4 shows the dependence of bisphere resonanceon detuning for TE30,1 and TM29,1 modes. Positionsof modes before contact are shown with dashed lines.Both TE30,1 and TM29,1 modes show anticrossing withsome symmetry. Such an asymmetry could be attributedto the influence of the broad second order ss ­ 2d WGM’s,and their positions are shown in the figure with thickshaded lines.FIG. 4. Resonance frequencies of coupled TE30,1 andTM29,1 for various bispheres with different size detuning,Dx ; 2pDayl. Dashed lines indicate the resonances ofuncoupled modes. Large open circles show the result of waveoptics calculation. Solid lines show the calculated normalmodes of bisphere as a function of size detuning using thecoupled harmonic oscillator model. Coupling constants areshown in Tables I and II.The observed features of intersphere coupling can beexplained by the resonance enhancement of the internalfield in bispheres. By expanding the internal and exter-nal fields of each sphere in terms of the vector sphericalharmonics under the plane wave excitation whose wavevector is parallel to the bisphere axis, we solve numeri-cally the coupled linear equation by the matrix inversionmethod [21]. The energy of the internal field is obtainedfrom the expansion coefficients as a function of the inci-dent plane wave parameters and sphere radii. Excellentnumerical convergence is achieved with the summation ofn up to 49. The spectrum shows prominent peaks asym-metrically located on both sides of TM29,1 and TE30,1modes. In between these prominent peaks there appearsa series of less intense peaks converging to the originalfrequencies of WGM’s. These substructures make theline shape asymmetric as we show in Figs. 2(b) and 2(c).Open circles of Fig. 4 show the positions of the promi-nent peaks. One may observe an excellent agreementbetween calculation and experiments from Fig. 4. Notethat both theory and experiment show asymmetry in themode splitting.In order to explore the feasibility of the tight-bindingscheme for the micromanipulation of light, it would bedesirable to introduce a mode overlap parameter whichcould describe the intersphere coupling. It is clear thatthese parameters will be determined by the convolution ofthe modes’ electric fields. Therefore, one can expect thestrong coupling for the modes with the matching frequen-cies whose orbitals lay nearly in the same plane and areclose to the contact point. We will refer coupling betweenthe modes with the same and different mode numbers as“intramode” and “intermode” coupling, respectively. Byintroducing the phenomenological coupling parameters ofthe dimension of frequency, which accounts for the inter-mode and intramode coupling for the relevant modes, wecan describe the bisphere fluorescence in standard termsof the interacting harmonic systems. In order to repro-duce the experimental results we take into account theintramode coupling between the WGM’s with s ­ 1, in-termode coupling of the WGM’s with s ­ 1 and s ­ 2and neglect the coupling between TE and TM modes. Thebest fit values of the coupling parameters are shown inTable I (TE mode) and Table II (TM mode); and the cor-responding detuning dependences are shown in Fig. 4 assolid lines.One can see from Fig. 4 that this model well ex-plains the overall features of the normal modes of bi-spheres and the asymmetric behavior of the upper modebranch of the TM29,1 due to the strong mixing with theTM25,2 mode. The best fit values of the coupling pa-rameters are almost constant for all the modes involvedin the narrow region of mode number variation fromn ­ 25 to 30.We perform four series of measurements using sphereswith diameters of 2, 3, 4.1, and 5 mm and estimate the4625VOLUME 82, NUMBER 23 P HY S I CA L REV I EW LE T T ER S 7 JUNE 1999TABLE I. Best fit values of coupling parameters (TE modes)normalized by resonance frequency.TE26, s ­ 2 TE30, s ­ 1 TE25, s ­ 2TE26, s ­ 2 3.4 3 1023 3.4 3 1023 · · ·TE30, s ­ 1 3.4 3 1023 3.3 3 1023 3.4 3 1023TE25, s ­ 2 · · · 3.4 3 1023 3.4 3 1023coupling parameters. The obtained coupling parametersnormalized to the resonance frequencies are 7 3 1023 for2 mm sn ­ 14d; 5 3 1023 for 3 mm sn ­ 21d; 4 3 1023for 4.1 mm sn ­ 29d; and 3 3 1023 for 5 mm sn ­ 36d.This implies that the coupling parameters decrease withincreasing sphere diameter or mode number. To explainsuch a behavior, one can recall that the smaller thespheres, the more the electric field outside the spheres,and hence the coupling parameter increases. One could bereminded here that the mode number represents the orderof the Hankel function which gives the field of WGMoutside the sphere. In the experiment two spheres arein contact and hence we obtain the maximum availablecoupling for a given size. The coupling can be reducedby increasing the distance between spheres as it has beendemonstrated for bigger spheres [14]. We would like toemphasize that the obtained coupling parameters are muchlarger than the line widths, indicating the dominance ofcoherent resonant coupling.In summary, we observe the coherent coupling ofWGM’s in bispheres. By the precise control of the spheresize, anticrossing behavior of the mode coupling is clearlydemonstrated. The features of normal mode splittingare perfectly reproduced by the wave optics calculations.Overall features of obtained bisphere normal modes arewell explained with the coupled harmonic oscillatormodel. The observed asymmetry in the splitting of upperand lower branches is ascribed to intermode mixing ofs ­ 1 mode with adjacent s ­ 2 modes. In the examinedsize region of spheres, the coupling parameters are foundto be much larger than the line width and smaller than themean mode separation of the WGM’s. This ensures thefeasibility of the tight binding manipulation of light wavesin a structure composed of contacting spheres. Thisscheme will also bring new insight into the applicationof WGM’s for micro-optical devices [22,23].TABLE II. Best fit values of coupling parameters (TMmodes) normalized by resonance frequency.TM25, s ­ 2 TM29, s ­ 1 TM24, s ­ 2TM25, s ­ 2 2.8 3 1023 3.5 3 1023 · · ·TM29, s ­ 1 3.5 3 1023 3.3 3 1023 3.5 3 1023TM24, s ­ 2 · · · 3.5 3 1023 2.8 3 10234626The author would like to acknowledge Dr. Yu. P.Svirko and Dr. C. Ramkumar for many fruitful discus-sions and Dr. Y. Li for her help during measurements.This work was supported by a grant-in-aid for ScientificResearch in the Priority Area “Near-Field Nano Optics.”*Author to whom correspondence should be addressed.Electronic address: gonokami@ap.t.u-tokyo.ac.jp[1] A. Mekis et al., Phys. Rev. Lett. 77, 3787–3789 (1996).[2] J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature(London) 386, 143–149 (1997).[3] J. D. 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Tight-Binding Photonic Molecule Modes of Resonant Bispheres

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Tight-Binding Photonic Molecule Modes of Resonant Bispheres

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