Fano resonances from coupled whispering – gallery modes in photonic molecules

whispering–gallery modes in photonic molecules THANH XUAN HOANG,1,* SARA NICOLE NAGELBERG,2 MATHIAS KOLLE,2 AND GEORGE BARBASTATHIS1,2 1Singapore–MIT Alliance for Research and Technology (SMART) Centre, Singapore 138602, Singapore 2Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA *hoangxuan11@gmail.com Abstract: We present a rigorous investigation of resonant coupling between microspheres based


Introduction
Coupling of resonant systems has attracted much attention in modern physics, both because it is useful in numerous applications and because observed behaviours are often elegant yet counterintuitive.The physical situations where such couplings occur are rather broadly distributed.The oldest and most celebrated analysis was presented by Fano in two papers separated by almost thirty years [1,2] in the context of photoionization.Fano's analysis refers to the transition of electrons from a discrete to a continuum state, which had been observed in earlier experiments [3] to exhibit a strongly asymmetric frequency response.He explained this transition as interference between two di erent paths from a discrete to a continuum state, one direct and another via a second discrete state; destructive interference between these two paths results in the observed asymmetric frequency response.Subsequently, so-called "Fano resonances" have been predicted and observed in semiconductor superlattices [4], Bose-Einstein condensates in optical lattices [5], atomic collisions (also known as Feshbach resonances) [6], and many other physical systems [7].Though Fano resonances are historically associated with asymmetric spectral profiles, it has been shown recently that symmetric profiles are allowed as well [7].Physically, Fano resonances are related to the interference between discrete and continuum states.In this paper, the spectral responses of this interference are referred to as Fano resonances and we show that we are able to engineer the Fano resonances by tuning the coupling strength between the states to obtain either symmetric or asymmetric spectral profiles.
In fact, asymmetric di raction from optical gratings [8] had been observed even earlier than photoionization, but the observation had remained theoretically unexplained until the development of Fano's theory.More recently, resonance coupling has been used to explain or design phenomena related to photonic structures such as coupled-resonator optical waveguides (CROWs) [9] and "photonic molecules" [10][11][12].Their purpose is generally to engineer frequency responses that would not be normally available from natural materials.For example, the CROW is a linear arrangement of weakly coupled sharply resonant cavities.By tuning the coupling, the dispersion of CROWs can result in small group velocity and, hence, slow light propagation [13,14].This e ect finds applications in quantum physics for quantum information technology [15] as well as biosensing and environmental sensing due to the sensitivity of the resonances to small perturbations [16][17][18].
Photonic molecules are a generalization of this idea.In actual molecules, the geometrical arrangement of atoms and interaction between their electron wavefunctions results in the overall frequency response; that, in turn, determines the spectral profiles of interactions between the atom's discrete states and continuum states.Photonic atoms and molecules are optically analogous: the "atoms" correspond to optical elements such as microrings, microdisks, microtoroids, nanoor microspheres, etc. [11,12].These are arranged geometrically such that their electromagnetic wavefunctions interact to produce desired spatial and spectral responses.For example, one can design arrays with extremely narrow directivity [11,19], or engineer the photonic molecules' coupled dielectric [20] or plasmonic [21] response to result in electromagnetically-induced transparency and tunability.
In this paper, we use the precise and e cient formulation of multipole eigenfunctions to obtain exact analytical predictions of scattering by photonic molecules.The same approach has already been followed in [22] for metal nanoparticles near plasmon resonance, based on the approach described by Gérardy [23].Multipole expansions have a rich and productive history [24], and are currently experiencing a renewed resurgence of interest [25][26][27].The advantage of using multipoles is that they provide exact and computationally manageable solutions to boundary value problems, especially involving clusters of spheres.Moreover, they permit treatment of arbitrary modes of illumination, including vector beams, e.g.radially polarized beams, vortex beams, etc.-as long as the illuminating radiation itself may be expanded in spherical harmonic functions [28,29].
In the paper by Sancho-Parramon and Bosch [22], the metallic particles were generally assumed to be of small diameter and, hence, their internal modes involved relatively few multipole modes.Therefore, the creation of dark states and Fano resonances required engineered illumination states: cylindrical vector beams, in particular.In the current work, we assume larger-diameter dielectric microspheres which do support optical whispering gallery modes (WGMs) [30] with narrower resonance bandwidth.When several microspheres are placed in close proximity to form a photonic molecule, complex interactions between their individual WGMs occur.We exploit this complexity to show that we are able to engineer Fano resonances with illumination by simple dipole radiation.
The structure of this paper is as follows: first, in section 2 we develop the multipole eigenfunction solution to the Maxwell's equations in a way that is compatible with general photonic molecule geometries; second, in section 3 we present the optical resonator-based coupling mechanism in the linearly coupled photonic molecules in which the spectral profiles of Fano resonance are discussed; we also discuss the waveguiding properties of coupled microsphere optical waveguides (CMOWs) in section 3.3 with focus on the influence of coupling strength on the bandwidths of the waveguides; next in section 4 we discuss light coupling and propagation in the photonic molecules with complex geometries; in section 5 we discuss the related topics of coupled nano-resonators and photonic crystals before concluding our paper in section 6.

Theory
This section is focused on the calculation of the electromagnetic fields in a photonic molecule due to its coupling with a local dipole source.As an example, Fig. 1 shows a T-shaped structure formed by five identical microspheres.A longitudinal electric dipole placed at the origin O acts as the light source throughout this paper.The electric field generated by the dipole is expressed in the coordinate system with axes x, z and origin O as follows: where p 1;0 is the moment of the electric dipole field N 1;0 and without loss of generality we set p 1;0 = 1.The dipole field is the outgoing electric multipole field N lm with the lowest order l = 1 and degree m = 0.More details about the definitions of the electromagnetic multipole fields can be found in our previous publication [31] and references therein.
The n-th microsphere has its own local coordinate system with the origin O n , which is translated from the O-coordinate system by a vector !OO n .The scattering field by the u-th microsphere is expressed in terms of electric N lm and magnetic M lm multipole fields as follows: where p (u) lm and q (u) lm are the electric and magnetic multipole moments, respectively.The number of multipoles included in the calculation is truncated by L u , which depends on the incident wave-number k and the radius R u as L u ⇡ k R u + 4.05 3  p k R u + 2 [32].The higher order multipole moments are negligibly small.Hence, we need to determine 2L u (L u + 2) unknown variables of p (u) lm and q (u) lm for calculating the scattered field E u ( ru ).This is accomplished by applying the following steps: The incident field approaching the n-th microsphere includes the scattered fields from all other microspheres Õ u,n E u ( ru ) and the originally incident field E 0 ( r): Next we express the left-hand side of Eq. ( 2) in the O n -coordinate system by applying the translational addition theorem [31].We then apply the boundary conditions for the n-th sphere to obtain where a (n) l 0 and b (n) l 0 are the Mie scattering coe cients resulting from applying the boundary conditions; the translational coe cients resulting from applying the translational addition theorem.It is clear that both magnitude and phase of the translational coe cients are relevant in the interference between the multipole moments.These coe cients hence play a role similar to that of the matrix elements in the original derivation of the Fano resonance [33].Equations ( 3) and ( 4) are valid for all l 0 and m 0 and hence we can form 2L n (L n +2) linear equations from applying the boundary conditions for the n-th microsphere.For a system of N coupled microspheres, we can form a system of L 0 = Õ N n=1 2L n (L n + 2) linear equations for determining the L 0 scattering multipole fields.This can be solved either by direct matrix inversion, or by the iterative biconjugate gradient method [25].The latter method is preferred for non-resonant cases since it is computationally cheap and has a fast convergence rate.In resonant cases, however, the iterative method is ine cient since the convergence rate is slow; therefore, direct matrix inversion is more e cient and, hence, was used in all results presented in this article.
Given the scattering multipole moments p (n) lm and q (n) lm resulting from the inversion, and the internal Mie scattering coe cients c (n) l and d (n) l , the multipole moments representing the internal field of the n-th microsphere are As we mentioned earlier, the truncation multipole order for the n-th microsphere is The term 3 p k R n accounts for the edge wave contributions associated with the high order multipole fields of order l such that k R n 4.05 3 p k R n + 2  l  L n [31].It is these higher order multipole fields that can accumulate su cient strength to resonate, and are referred to as whispering gallery modes (WGMs) of the photonic molecules.Their corresponding multipole moments ⇣ (n) lm and ⌘ (n) lm describe the WGMs' spatial and spectral structures, taking correctly into account the coupling between the fields in the individual photonic atoms (individual spheres,) in the atoms' interiors.For a single isolated photonic atom this would not have been necessary; instead, the Mie scattering coe cients a (n)  l and b (n) l in the atom's exterior and c (n) l and d (n)  l in the interior would have su ced [31].
Before we close this section, it will be useful to define the quality factor of the (n) plm -th WGM as where g is the radial mode number; and (n) glm and (n) glm are this (n)  glm -th mode's resonant wavelength and full width at half maximum of the spectral profile, respectively.These are obtained directly from plotting the magnitudes of the multipole moments ⇣ (n)  lm or ⌘ (n) lm for the electric or magnetic multipoles, respectively.

Linear chains of optically coupled microspheres
As an example, we study the optical properties of several photonic molecules, such as the one shown in Fig. 1, formed by identical flint-glass microspheres whose radius and dielectric constant are R = 1.5 µm and " = 3.24, respectively [34] for the visibly spectral band 0.4 µm   0.7 µm.With these parameters, the appropriate truncation for the multipole order is L ⇡ 37.

Single photonic atom
First, let us study the optical properties of a single (isolated) photonic atom, i.e., we study the microsphere denoted as O 1 in Fig. 1 in absence of the other microspheres.For this isolated atom, only the coe cients ⇣ (1)  lm with m = 0 are non-zero; the magnetic multipole fields are also all identically zero in this case.Therefore, we may express the internal field in terms of L = 37 regular electric multipole fields N m l ( r) with m = 0 as follows: ⇣ (1)  l;0 N 0 l (r 1 ).
Moreover, because the modes are orthogonal, there is no coupling between them and, hence, no Fano-Feshbach resonances are to be expected.We need more significant digits for determining Figures 2(a) and 2(b) show the spectral profiles of the two multipole moments ⇣ (1)  l;0 with l = 25 and l = 33, respectively, chosen because they have an overlapping resonance at (1)  1;33;0 .This overlapping resonance accounts for the interesting e ect of coupled-resonator-induced transparency [7,20].The other 35 multipole moments are continua around (1)  1;33;0 .The exact values of the highest resonances of the two WGMs are (1) 2;25;0 = 0.430 646 000 00 µm and (1) 1;33;0 = 0.431 472 568 42 µm.For brevity, we refer to (1)  1;33;0 as 0 hereafter.The 25-th multipole has an additional resonance at (1)  1;25;0 = 0.478 787 000 00 µm, which is too far from 0 to contribute to the spectral responses of the photonic molecules around 0 .It is necessary to determine the strongest resonances with the significant digits since we are studying the modes with ultra-high quality factors.This requirement of keeping the high precision can be relaxed for modes with lower quality factors.In our current case, the quality factors of the two WGMs are found to be Q 25 ⇡ 290 for the weak mode and Q 33 = 1.679 ⇥ 10 8 for the strong mode.As expected, the independently resonant modes have Lorentzian spectral profiles; and at the 0 resonant wavelength, the field is dominated by the 33-rd multipole as shown in Figs.2(c) and 2(d).
3.2.Linearly-coupled microsphere molecules Next, we study the optical coupling between the two photonic atoms O 1 and O 2 in intimate contact as shown in Fig. 3. Similarly to Eq. ( 7), the internal electric field of the second atom O 2 may be expressed as Using Eqs. ( 3) and ( 5), we can express the coupling between the two internal fields as follows: The summation Õ 37 l=1 in Eq. ( 9) represents the interference between the strongly (l = 33), weakly (l = 25), and the other 35 radiative modes at 0 .In the context of the Fano-Feshbach resonance of overlapping resonances [7], the weak and strong modes play the roles of the probe and pump beams, respectively.
Figure 4 shows the asymmetric spectral profiles of the two multipole moments ⇣ 25;0 and ⇣ 33;0 .Similarly to the spectrum of the probe beam in the Fano-Feshbach resonance, the spectrum of ⇣ 25;0 is enhanced and suppressed before and after the wavelength 0 as shown in Fig. 4(a).Figure 4(b) shows the spectral shift and split of the WGM ⇣ 33;0 due to the strong coupling between the two microspheres.In fact, the spectral shift and split of two-sphere systems were predicted earlier in references [35,36] in which the authors studied the forward-scattered intensity [35] and the internal energy [36] of the microspheres under plane-wave illumination.Here, we present a novel approach based on studying the coupling between multipole fields in the systems under local-source illumination.Our approach has two main advantages.Firstly, the spectral profiles of the multipole moments, which directly correspond to the resonant and radiative modes, enable us to quantify the resonant strength and radiative loss in the coupled systems.Secondly, using the local-source illumination instead of using the plane-wave illumination o ers a more accurate explanation of the optical coupling between the microspheres.Intuitively, we observe from Fig. 3 that the optical field in the 2-nd microsphere is mainly due to the scattering field from the 1-st microsphere.Now, assuming we replace the dipole source by plane-wave illumination, due to the di raction nature of light, the internal field of the 2-nd microsphere results not only from the scattering field of the 1-st microsphere but also from the di raction component of the incident plane wave.To be clearer, let us consider the case of increasing d !+1: for the local-source illumination, the internal field of the 2-nd microsphere will be nearly zero, since there is no coupling between the two microspheres; for the plane-wave illumination, the internal fields in the two microspheres will be exactly the same regardless of the direction of the illumination, since the incident plane wave is exactly the same for both of the microspheres.To mathematically explain the spectral responses in Fig. 4, we consider the coupling between the two internal multipole fields.As suggested by Eq. ( 9), the resonant mode l = 33 in one microsphere is coupled to all other modes including the resonant l = 33 and non-resonant l , 33 modes in the other microsphere.These non-resonant modes are usually referred to as the radiative modes and account for the system's loss.This loss is the reason why the quality factor decreases by four orders of magnitude from , the Q-factor of the stronger mode in Fig. 4(b).The quick transitions of the phases presented in Fig. 4 correspond to the resonant enhancement or anti-resonant suppression wavelengths.
Figure 5 shows how the Fano resonance can be engineered by tuning the gap distance between the two identical microspheres.Increasing the gap distance to d = d 2R = 100 nm in Fig. 5(a), the quality factor of the first peak increases from Q ⇡ 1.78 ⇥ 10 4  to Q ⇡ 9.5 ⇥ 10 4 .This Q-factor increase is due to the fact that the loss decreases in the system when we weaken the coupling between the resonant and radiative modes by increasing the gap distance., respectively.We emphasize that the splits are not of the Autler-Townes type [37].The latter occurs in the presence of a strong pumping beam, when an atomic state is split into two dressed lifetime-broadened states [38].Our case is di erent in that there is no  pumping beam; however, the dipole wave can be thought of as the probe beam.The splits are due to the coupling between the two internal fields as pointed out by [20].Another interesting observation from Fig. 5(b) is that the spectral profile of the Fano resonance becomes more symmetrical for the large gap distance.Increasing the gap distance further to 1 µm, results in the two peaks merging into a single peak, as observed in Fig. 5(c).Since the coupling is weaker, the loss is also smaller and, hence, the resonances become stronger in both microspheres.For the gap distance of 2 µm in Fig. 5(d), the quality factors of the WGM in the two microspheres are Q ⇡ 1.547 ⇥ 10 8 and Q ⇡ 2.6967 ⇥ 10 8 , respectively.The first microsphere is e ectively independent from the second microsphere.The presence of the first microsphere boosts the quality factor of the second microsphere in the weak coupling case.This property may be useful for designing optical filter and delay lines.However, due to the weak coupling, the resonant strength in the second microsphere is several times weaker than that in the first microsphere as shown in Fig. 5(d).In fact, the dependence of the Q-factor on gap distance was studied earlier for coupled microdisks [39,40] and microspheres [41], in which the authors used di erent numerical approaches.Our multipole-based approach o ers an alternative e cient spectral analysis tool for designing the spectral responses of photonic molecules.Here, our approach is used to show that we are able to engineer the Q-factor, shape, and spectral split of the Fano resonance by tuning the gap distance.
Next, we consider three linearly coupled microspheres, as shown in Fig. 6. Figure 7(a) shows the magnitude of the multipole coe cient ⇣ 33;0 exhibiting three peaks.Coupling the third microsphere into the two-coupled-microsphere chain results in adding more resonant and radiative modes into the interference in Eq. (9).Consequently, in comparison with the magnitude plot in Fig. 4(b), one more peak appears in Fig. 7(a).The two stronger peaks in Fig. 7(a) are centered at 1 = 0.430 992 µm and 2 = 0.431 686 µm.The resonance in the second microsphere is suppressed at 2 because the coupled fields from the first and third microspheres interfere destructively.This interesting case presents an analogy with the nodal plane concept in molecular physics.The middle microsphere functions as "nodal" in this case, in the sense that the atomic wave-functions from the edge spheres interfere destructively at the location of the middle sphere.Interestingly, the center mode was experimentally observed earlier to be dominant for the first and third microspheres in reference [42], in which the authors used a tight binding model for numerically predicting the symmetrical spectral split.On the contrary, our model results in the strongly pronounced asymmetric split as shown in Fig. 7(a).The di erence between these two models is because the tight binding model neglected the contributions of the radiative modes.However, this neglect is justified only in weakly coupled systems as discussed in [9].For the weakly coupled microspheres in Figs.5(b) and 5(c), our model predicts the nearly symmetric splits in agreement with those that would be predicted by the tight binding model.Figures 7(b) and 7(c) show the spatial electric intensity distributions for the two resonant wavelengths 1 and 2 , respectively.As expected, the field resonates in all the three microspheres at wavelength 1 .At wavelength 2 , the middle microsphere in Fig. 7(c) is nearly o -resonance.Figures 5 and 7 show that the resonant strengths in the first and last microspheres are the same in the strong coupling regimes.In other words, the WGM can e ciently propagate through the coupled-microsphere systems.

Linearly-coupled microsphere waveguides
The concepts of CROW and photonic molecules appeared almost at the same time [9,10].CROWs were initially proposed for the possiblility of highly e cient nonlinear optical frequency conversion and perfect transmission through bends.Since then CROWs have garnered great interest from both theoretical [34,43] and experimental [16,[44][45][46] researchers.In the original article, Yariv et al. proposed CROWs formed by weakly coupled micro-disks, which support high-Q WGMs.The weak coupling condition is necessary to ensure that the spectral profile of the WGM in each constituent resonator remains essentially the same when coupled with other resonators.This condition allowed the authors to formulate the waveguiding mode in the context of the tight-binding approximation, which is extensively used in the solid state physics.In this tight binding model, the waveguiding mode is the combination of the WGMs of all the resonators, hence this model neglected the coupling between the high-Q WGMs and radiative modes.The validity of the weak coupling condition for the two coupled microspheres can be appreciated from Fig. 5.The resonant wavelength of the coupled microspheres is essentially the same as that of the single microsphere only in Fig. 5(d) with the gap distance of d = 2 µm (about 5 0 ).For the stronger-coupling cases in Figs.5(a)-5(c), the resonant wavelengths are significantly di erent from that of the single microsphere and hence the tight binding model is not applicable to these stronger-coupling systems.Hence, our multipole-based approach has the advantage of being    applicable to both weakly and strongly coupled systems.Figure 8 shows the spectral profiles of the multipole moments ⇣ 33;0 representing the WGMs in the first two and last three microspheres of a CMOW formed by 50 microspheres in pairwise contact.In analogy to one-dimensional photonic crystals [47], the CMOW also has a bandwidth.In our current case, the bandwidth is ⇡ 1 nm.With the wavelength in the pass-band, the resonant strengths in the first and last microspheres of the CMOW are comparable.Similar to the suppression presented in Fig. 7(a), the resonant strengths in the 1-st, 2-nd microspheres in Fig. 8(a) and the 48-th and 49-th microspheres in Fig. 8(b) can be suppressed within the pass-band.For wavelengths in the stop-band, the resonant strength in the last microsphere is nearly zero.The resonant wavelength 3 in Fig. 8(a) is used for obtaining the electric intensity distributions in Fig. 9.The spatial distributions in the first and the last microspheres are nearly identical.In fact, the distributions in the first 25 and last 25 microspheres are similar though the light source is placed at the 1-st microsphere.To understand the importance of the radiative modes and WGMs, the multipole moments representing the modes of the 1-st and 50-th microspheres are shown in Figs.10(a) and 10(b), respectively.The scattering multipole moments p l;0 of the radiative modes (l  26) are significantly higher than that of the WGMs with l = 33.Hence the radiative loss of the waveguide is mainly due to the coupling between the WGMs with the radiative modes.As expected, the internal multipole moments ⇣ 33;0 of the WGMs dominate that of the other radiative modes l , 33 inside the microspheres.Hence, these WGMs may be referred to as dark modes in the sense that their scattering moments are significantly weaker than that of other radiative modes.To observe the e ects of the gap between the microspheres on the waveguiding properties, we increase the gap distance.Figures 11(a  microspheres are still comparable.When increasing the gap distance further to d = 800 nm, the optical coupling in the CMOW is su ciently weak that the resonance in the last microsphere is much weaker than that in the first microsphere as observed in Fig. 11(c).
Figure 12 shows the spectral profiles of the WGMs in the first two and last three microspheres of the CMOW formed by the 100 touching microspheres.Figures 8 and 12 show that the bandwidths of the two CMOWs are almost the same.E ectively, we may consider both of the CMOWs as infinite CMOWs with the same bandwidth of ⇡ 1 nm.More peaks appearing in Fig. 12 is a result of adding more modes into the interference.The quality factors of the peaks in Fig. 12 are higher than that of the peaks in Fig. 8. Nevertheless, the quality factors corresponding to these peaks are always lower than that of the independent microsphere due to the coupling to the radiative modes.We have shown that the waveguiding properties of coupled-microsphere optical waveguides can be engineered by tuning the gap distance, e.g., the shape of the pass-band and the quality factors of the resonant peaks in the pass-band can be designed by choosing an appropriate gap distance.These waveguiding properties are important for some applications, such as narrow-line filters and delay lines.Regarding these practical applications, it is worth discussing possible e ects of spectral detuning between individual microspheres, which is caused by inevitable variations in size and shape of the microspheres.These detuning e ects were experimentally studied in references [48,49].Theoretically, the detuning e ects were analyzed by using finite di erence time domain software [41], a tight binding model [42], and an approximate multipole-based model [50].Experimentally, Astratov and co-workers reported the best intensity attenuation of 3 to 4 dB per sphere for chains of touching polystyrene microspheres with sizes in the 3-20 µm range and a size dispersion of 1% [48].Both the theoretical and experimental works showed that detuning deteriorated the waveguiding e ciency significantly.However, our results presented in Figs. 8 and 12 suggest that it is possible for the resonant strengths in the first and last microspheres to be the same even with spectral detuning.Since the bandwidths of the touching microspheres are as large as 1 nm, we expect that spectral detuning within 1 nm still results in the same resonant strengths.Small detuning may be obtained, for an example, by using a sorting technique based on the resonant light force [51].Though our model is readily applicable for studying these detuning e ects, analyzing them in further detail would go beyond the topic of this paper.

Complex photonic molecules
For the simple molecules in section 3, only the WGM with the azimuthal mode number m = 0 gets excited in the system.In general configurations of photonic molecules, all of the WGMs with the polar l = 33 and azimuthal 33  m  33 mode numbers get excited in the molecules.The spectral profiles of the WGMs are in general di erent and depend on the molecule geometries.We may substitute Eq. ( 5) into Eq.( 4) for obtaining the coupling between the wavefunctions of the constituent atoms.Figure 13 shows the simulation results for the three molecules M, I, and T. First, we discuss the result for the M molecule.The resonance in the M molecule is much stronger than that in the I and T molecules.This stronger resonance is mainly due to the gap distances between the microspheres.To understand this strong resonance in further detail, we consider the two optical paths of the resonant light in Fig. 2(c).The light circulating along path 1 in each microsphere is weakly coupled to the other microspheres due to the large gap distances (measured in the corresponding planes) between them.Consequently, the resonance along path 1 is strong as indicated by both the spectral profiles and the electric intensity distribution in Fig. 13(a).The light circulating along path 2 is strongly coupled to the other microspheres and hence the resonance along path 2 is weak as observed from the electric intensity distribution in Fig. 13(a).Another noteworthy property of the WGMs is related to their degeneracy.It has been known that for the single independent microsphere, all the WGMs with l = 33 and 33  m  33 are degenerate, i.e., they have the same spectral profiles.This degeneracy is lifted in the coupled microspheres since the coupling strengths between the microspheres depend on the optical paths, i.e., depend on the molecule geometry.For the linear chain in Fig. 9 Here, it is important to note that the definition of the azimuthal mode number m must be always associated with a specific coordinate system.Each azimuthal number corresponds to one resonant mode in the coordinate system as can be visualized in reference [52].When describing this resonant mode in other coordinate systems, we need to use rotational and translational addition theorems for expressing it in terms of new eigenfunctions with di erent azimuthal mode numbers.In general, we need all of the resonant azimuthal modes with l  m  l for describing one resonance.For an example, all of the WGMs with 33  m  33 contribute to the resonances in Fig. 13.Due to this fact, we believe that to explain numerical and practical observations, the resonant planes [52] and equivalently the resonant paths in this paper are more appropriate than any single azimuthal resonant mode.
The resonances in Figs.13(b) and 13(c) are weaker than that in Fig. 13(a).The reason for these weaker resonances is the strong coupling between the 1-st and 2-nd microspheres for all the optical paths including path 1 and path 2 in Fig. 2 We now study the so-called super-resonator [9]   .For this particular wavelength 5 , each of the three identical smaller microspheres acts like a lens, whereas the big microsphere acts like both a resonator and a lens.
Optically coupled microspheres have originally attracted researchers for exploiting their resonancebased ability in storing and molding the flow of light.Recently, these coupled systems have also gained considerable attention from researchers for exploiting their o -resonance-based ability in focusing and guiding light [44,45].In this section, we distinguish and discuss two optical coupling mechanisms in the on-and o -resonance systems.These two mechanisms are shown in Fig. 15.The bigger photonic atom has the radius of R = 2.37 µm and the dielectric constant of regular glass " = 1.46 2  .One resonant wavelength of the atom is 5 = 0.439 44 µm [31], which is used for obtaining the electric intensity distribution in Fig. 15.For the three o -resonant microspheres, the optical coupling is mainly due to the lensing e ect of the microspheres.Each of these microspheres acts like a lens, which has the focus at exactly its back surface [34].Hence, the light radiated by the dipole at point A is approximately collimated by the first lens and then is focused into the point B by the second lens.Since all the three small microspheres are o -resonance, the optical coupling is due to the radiative modes and hence the radiative loss is high.This lensing e ect is important in explaining the formation of the periodically focused beam along the linear arrays of the microspheres in [46].The second optical coupling mechanism involves the WGM.The light scattered by the three small microspheres approaching the bigger microsphere can be categorized into two parts.The first part approaches the microsphere's edge and circulates resonantly along its circumference to form the WGM.This optical path is represented by the white circle in Fig. 15.The second part does not approach the edge and hence get reflected and refracted by the bigger microsphere.This optical path is represented by the dash white arrow in Fig. 15.Hence, the big microsphere acts like both a resonator and a lens.In general, both resonant and radiative modes are important in explaining the optical coupling mechanisms and the Fano resonances resulting from the coupling between these modes should not be neglected in studying the spectral responses of coupled resonant systems.
In molecules with arbitrary shapes, the optical coupling based on the lensing e ect always exists.Meanwhile, the resonance coupling occurs only when at least one mode of the excitation source matches with one mode of the molecules.Therefore, to accurately explain a practical spectral response of a molecule, it is necessary to reveal both its molecular modes and excitation modes.The latter can be selected -for an example -by tailoring a light beam.The molecular modes of an arbitrary molecule is in general highly complicated.To simplify our discussion, we choose a molecule comprising N identical atoms.We showed in section 4.1 that for each polar resonant-mode number l, each atom in the molecule supports 2l + 1 non-degenerate azimuthal modes.When it is excited, each of these azimuthal modes couples with all of the (N 1)(2l + 1) resonant and (N 1)(2L(L + 2) 2l 1) radiative modes of the other atoms to result in a collective spectral response of the molecule.This coupling strongly depends on the molecule's spatial configuration, as we discussed earlier.Consequently, the practical spectral behavior of the molecule strongly depend on both its spatial configuration and the external excitation.

Discussion
Light coupling and propagation in systems of nano-and micro-particles have been of great interest for the last several decades.These nano-and micro-particles are usually referred to as nano-and micro-resonators.Metallic nano-particles, which can confine and guide light at the nanoscale, are of great interest for nanophotonic applications, since they support localized surface plasmon resonances.However, these localized surface plasmon resonances are always associated with significant energy dissipation and hence a reduction in the device e ciency.Due to these limitations, high-refractive-index dielectric nanoresonators have recently emerged as alternative buiding blocks for photonic devices [27,53].This shift of interest is founded in the ability of high-refractive-index nanoresonators to support both low-order electric and magnetic multipole resonances, i.e., low-Q WGMs.The spectral overlap between these broad resonant modes spans through the whole visible wavelength range.Tailoring of excitation beams by focusing polarized light [28,29] and exploiting the rich spectral properties of the dielectric nanoresonators lead to many interesting phenomena, such as selectively multipole excitation [54] and strong lateral directionality [55].Clusters of both metallic [22] and dielectric [56] nano-resonators have been extensively studied in the context of the Fano resonance for both far-field and near-field property enhancements, which find various applications, such as metasurfaces or light management in photovoltaic systems [56,57].Our spectral analysis approach presented in this paper is readily applicable to these coupled systems.Studying the spectral profiles of individual multipole moments enables us to categorize the corresponding multipole fields into resonant or radiative modes.We are then able to tune the interference between these modes by arranging the resonators' positions.Through this tuning of the Fano resonance, the collective spectral responses of the systems can be analyzed and optimized for practical applications.
Light localization in a two-or three-dimensional photonic crystal, formed from a periodic arrangement of nano-or micro-particles, is entirely due to coherent multiple scattering and constructive interference of light from the constituent particles [47].Similarly, a WGM is due to constructive interference between multiple scattering occurring inside the corresponding microsphere.This multiple scattering is represented by the Debye series as explained in [58].The localization of light is analogous to an excited discrete atomic state and hence results in the Fano resonance observed in its interference with other optical continuum states [7].This interference is physically analogous to that between the resonant and radiative modes in the coupled microsphere sytems.Due to this analogy, we expect that our results presented in this paper may shed some light on the light propagation in photonic crystals.

Conclusion
We present a spectral analysis approach for studying systems of optically coupled nano-and micro-spheres.Our idea is to study the spectral profiles of the multipole moments of the resonant and radiative modes, which correspond respectively to the resonant strengths and radiative loss in the systems.The spectral responses of the interference between these modes are referred to as Fano resonances in this paper.Both symmetric and asymmetric spectral profiles are shown to be allowed in these Fano resonances.We also show that the quality factor and shape of the Fano resonances can be engineered by arranging the relative positions of constituent microspheres.In the strongly coupled microspheres, our model results in the strongly asymmetric spectral splits that indicate the crucial roles of both the resonant and radiative modes in analyzing the spectral responses.Our approach may also be useful for studying the spectral responses of photonic molecules in the framework of atomic and molecular physics.Analyzing the spectral responses in this direction may o er new insights into molding the flow of light by using systems of coupled resonators.

Fig. 1 .
Fig. 1.The T-shaped molecule is formed by five identical microspheres.The local source is an electric dipole placed at the origin O and pointed along the z direction.The dipole-molecule distance D determines the coupling strength between the local source and the molecule.The distance d between the microspheres determines the overlap between the wavefunctions of two adjacent microspheres, and hence determines the optical coupling strength in the molecule.

Fig. 2 .
Fig. 2. Spatial and spectral responses of a single photonic atom.(a) The magnitude of ⇣ (1) 25;0 ;0 since its associated whispering gallery mode has a higher quality factor in comparison with those of show the quick transitions at the resonant wavelengths.(c) Spatial distribution of the electric field intensity with the 33rd mode in resonance at excitation wavelength (1) 1;33;0 .Paths 1 and 2 show the two propagation paths of the resonant light on the two orthogonal planes of x = 0 and y = 0, respectively.(d) The electric field intensity distribution in logarithmic scale for better visualization.

Fig. 3 .
Fig. 3. Two linearly coupled microspheres.The gap distance ( d = d 2R) determines the spectral behavior of the molecule.

Fig. 4 .
Fig. 4. Spectral profiles of the multipole moments of the photonic molecule formed by the two touching microspheres.(a) The magnitude and phase plots of the multipole moments ⇣ 25;0 .(b) The magnitude and phase plots of the multipole moments ⇣ 33;0 show two resonant peaks.

Fig. 5 .
Fig. 5. E ects of the gap distance d = d 2R on the spectral profiles of the multipole moments ⇣ 33;0 .(a) The two resonant peaks are stronger and closer to each other in comparison with the two peaks in Fig. 4(b).(b) The two resonant peaks get closer to the central resonant wavelength 0 .The unit of the transversal axis corresponds to 10 7 µm.(c) The two peaks merge into one centered at 0 .(d) The weak coupling results in the weaker resonance in the second microsphere and no spectral split.

Fig. 7 .
Fig. 7. (a) The spectral profiles show three resonant peaks including the highest peak at 1 = 0.430 992 µm and the second highest peak at 2 = 0.431 686 µm.(b) The electric intensity in logarithmic scale shows that the light with the wavelength 1 resonates in all three microspheres.(c) For the wavelength 2 , the resonance in the middle microsphere is suppressed.

Fig. 8 .
Fig. 8. (a) The strongest resonances in the first and last microspheres occuring at 3 = 0.431 393 µm have the same strengths.(b) Magnitudes of the multipole moments in the last three microspheres show that the resonances in the 48-th and 49-th microspheres are suppressed at di erent wavelengths in the bandwidth.This means that the resonant strengths are not the same in all of the constituent microspheres.

Fig. 9 .
Fig. 9. Electric intensity distributions in logarithmic scale with 3 = 0.431 393 µm for (a) the first five microspheres and (b) the last five microspheres.The distributions between the first and last microspheres are nearly identical though the light source is placed at the first microsphere.

Fig. 10 .
Fig. 10.Multipole moments represent the scattering and internal fields of the first and last microspheres.(a) The scattering moments of the resonant modes l = 33 are negligible in comparison with the radiative modes 1  l  26.(b) The moments of the resonant modes dominate that of the other modes.

|Fig. 11 .
Fig. 11.E ects of the gap distance on the spectral profiles of ⇣ 33;0 .(a) The bandwidth decreases to = 0.26 nm.(b) The resonances in the first and last microsphere are still comparable.The bandwidth decreases to as narrow as = 1.66 x10 7 nm.(c) The resonance in the last microsphere is weaker than that in the first microsphere.
)-11(c) show the spectral profiles corresponding to the gap distances of d = 100 nm, d = 700 nm, and d = 800 nm, respectively.Increasing the gap distance d reduces the bandwidth of the CMOW.As observed in Fig. 8, the bandwidth and the strongest resonant strength for the touching CMOW are 1 nm and |⇣ 33;0 | = 12.52, respectively.Increasing the gap distance between the microspheres to d = 100 nm reduces the bandwidth from = 1 nm to = 0.26 nm and enhances the strongest resonant strength from |⇣ 33;0 | = 12.52 to |⇣ 33;0 | = 120, as shown in Fig. 11(a).
Figure 11(b) presents the spectral profiles for the CMOW with the largest gap ( d = 700 nm), at which the resonances in the first and last

|Fig. 13 .
Fig. 13.Light resonates in the complex molecules.The electric intensity distributions are plotted using the strongest resonance wavelengths of the corresponding molecules.(a) The 4-th microsphere is placed at x = 1.5 µm, y = 0 µm, and z = 4.8 µm.Due to the molecule geometry and the gap between the 4-th and other microspheres, the light paths 1 (on the plane x = 0 as explained in Fig. 2(c)) of the microspheres are relatively independent.This independence explains the strong resonance in the M molecule.(b) The strongest moments in the 6-th, 5-th, and 7-th microspheres are roughly half of those of the 1-st, 2-nd, 3-rd microspheres, respectively.Intuitively, this is due to the fact that light circulating along path 1 in the 1-st, 2-nd, 3-rd microspheres is not much a ected by the presence of the 4-th microsphere.(c) In comparison with the I molecule, the multipole moment in the 5-th is much weaker for the T molecule due to the absence of the 6-th and 7-th microspheres.
, only the WGM with l = 33 and m = 0 contributes to the resonance.For the M molecule, all the WGMs with l = 33 and 33  m  33 contribute to the resonance.The spectral profiles in Fig. 13(a) represent the strongest modes in the studied wavelength range.The strongest mode in the 1-st, 6-th, and 7-th microspheres is the same WGM with l = 33 and m = 0.The strongest modes (l = 33) in the 2-nd, 3-rd, 4-th, and 5-th microspheres correspond to m = 2, m = 8, m = 24, and m = 22, respectively.The weaker modes are not included in the spectral plot.

Fig. 14 .
Fig. 14.Super-resonator: Adding more microspheres increase optical couplings between the resonant and radiative modes, i.e., increasing the radiative loss.Consequently, the resonance is not enhanced by forming the resonator of resonators.(a) The spectral profile.(b) The electric intensity distribution in linear scale.(c) The electric intensity distribution in logarithmic scale with 4 = 0.431 µm.
by folding the photonic molecule back upon itself as shown in Fig.14.The formation of a resonator from resonators is aimed at trapping the resonant light for a longer time by forming a closed loop.On the contrary, our result shows that forming a closed loop does not enhance the Q-factor of the resonant mode in comparison with an open loop.The resonances representing the 1-st, 2-nd, and 3-rd microspheres in Fig.14(a) are similar to those for the I molecule as shown in Fig. 13(b).These similar resonances are mainly because the light circulating along the optical path 1 is not a ected by the presence of the 4-th and 8-th microspheres.The resonances in the other five microspheres are due to the light circulating in the optical path 2. The impact of the presence of the 4-th and 8-th microspheres on the spectral responses in the 1-st, 2-nd, and 3-rd microspheres may be thought of as that of the substrate presented in reference [52].The resonances in the 5-th, 6-th, and 7-th microspheres are not enhanced by forming a closed cloop.This is simply because adding more microspheres into the loop results in more coupling between the resonant and radiative modes and consequently results in more radiative loss.This additional radiative loss ultimately prevents any Q-factor enhancement of the super-resonator.The electric intensity distribution in Figs.14(b)-14(c) is obtained for the wavelength of 4 = 0.431 µm.Another interesting observation from Fig. 14(a) and Fig. 13(b) is related to the strongly asymmetric spectra of the Fano resonances in the 5-th, 6-th, and 7-th microspheres around 4 .spectral property may be exploited for optical switching technology, in which the resonances in the 5-th, 6-th, and 7-th microspheres can be switched on and o by slightly tuning the illumination wavelength.

Fig. 15 .
Fig. 15. 5 = 0.439 44 µm is the resonant wavelength of the big photonic atom with radius R = 2.37 µm and dielectric constant " = 1.46 2. For this particular wavelength 5 , each of the three identical smaller microspheres acts like a lens, whereas the big microsphere acts like both a resonator and a lens.