The optical coupling between two size-mismatched spheres was studied by using one sphere as a local source of light with whispering gallery modes (WGMs) and detecting the intensity of the light scattered by a second sphere playing the part of a receiver of electromagnetic energy. We developed techniques to control inter-cavity gap sizes between microspheres with ~30nm accuracy. We demonstrate high efficiencies (up to 0.2–0.3) of coupling between two separated cavities with strongly detuned eigenstates. At small separations (<1 µm) between the spheres, the mechanism of coupling is interpreted in terms of the Fano resonance between discrete level (true WGMs excited in a source sphere) and a continuum of “quasi”-WGMs with distorted shape which can be induced in the receiving sphere. At larger separations the spectra detected from the receiving sphere originate from scattering of the radiative modes.
©2006 Optical Society of America
The mechanisms of optical coupling between microcavities supporting high quality (Q) whispering gallery mode (WGM) resonances have attracted considerable interest in the past few years. This interest is driven by potential applications of integrated microrings, cylinders, or spheres for controlling dispersion relations for photons in various structures such as high order filters , coupled resonator optical waveguides [2–4] (CROW), side-coupled integrated spaced sequences of resonators  (SCISSOR) and more complicated [6–11] coupled cavity structures. In most of the theoretical work on this subject the cavities are assumed to be identical, leading to efficient resonant coupling between WGMs in adjacent cavities. Such coupling has an analogy with a coherent interaction between atoms  in a quantum limit.
It should be noted however, that the studies of coupling mechanisms between nonidentical cavities constitute an important subject for developing applications of multiple cavity systems. Firstly, the detuning between the cavities can be introduced in a controllable fashion to achieve vernier effect tuning , stabilize mode-locking  of lasers or to expand the transmission band of such filters and delay lines. Secondly, continuous improvement of Q factors of individual cavities due to technological breakthroughs makes it inevitable that even systems with highly uniform resonators begin to display the properties of a disordered system if the random detuning between the cavity resonances exceeds their individual linewidths.
Dielectric microspheres can be considered as very attractive building blocks of more complicated coupled cavity structures due to ultra-high Q factors of their WGM resonances, and due to the possibility of sorting cavities on the basis of their spectral characterization. Some effects determined by the interplay of order and disorder in such systems have already been observed. These include normal mode splitting [15–17] and band structure effects [18,19] in uniform chains, as well as optical transport effects  in systems with disorder. The coupling between detuned cavities is, however, not well studied at present.
Recently it has been argued that coupling between cylindrical or spherical cavities can be achieved by two mechanisms distinctly different from the resonant optical tunneling. In one of these mechanisms  the coupling occurs due to a Fano resonance between a discrete energy state (true WGM) in a sphere containing a source of light and a continuum of “quasi”-WGMs with irregular shape which can be induced in the second sphere. Such evanescent coupling can be achieved between the cavities with strongly detuned eigenstates in a regime of weak coupling. Another mechanism is based on the phenomenon of formation of photonic nanojets  emerging from the shadow-side surface of a dielectric microcylinder or a microsphere under plane wave illumination. In a chain of nanojet-inducing microspheres, the constituent spheres are coupled  due to the periodical nature of nanojets.
In the present work by employing scattering spectroscopy with spatial resolution we demonstrate high efficiencies (up to 0.2–0.3) of coupling between two separated, size-mismatched cavities with strongly detuned WGM eigenstates. We developed techniques for controlling interresonator gap sizes between optically coupled microspheres with nanometric (~30 nm) accuracy. At small inter-cavity separations d<1 µm, the coupling is shown to have an evanescent nature, and is explained by the recently suggested mechanism  of Fano resonance between true WGMs excited in the cavity containing a source of light, and a continuum of quasi-WGM states which can be induced in the cavity receiving electromagnetic energy. At larger separations the spectral peaks detected from the receiving sphere are explained by the illumination effect provided by the scattering of WGMs in the source sphere.
2. Structures and method of controlling the inter-cavity gap sizes
We study optical coupling phenomena in dielectric bispheres by using one of the spheres as a local light source (S) and the second sphere as a passive radiation receiver (R). The sizes of the spheres were different in most of the experiments. The source of light was created by dye-doped fluorescent (FL) polystyrene microspheres with sizes D s=10 or 5 µm and size dispersion ~3% obtained from Duke Scientific Corp. As the receivers of radiation we used undoped spheres with various sizes in a 3 µm≤D r≤20 µm range and size dispersion ~1%.
We developed a technique for controlling the separations between the cavities based on placing the spheres at the top of the stretchable substrate, as illustrated in Figs. 1(a) and 1(b). If the spheres are attached to the substrate and not to each other, one can control the separation (d) between the spheres by creating a tensile strain in the substrate. The substrate material was selected on the basis of its ability to withstand a significant strain up to ~0.2, and to provide strong surface adhesion with the spheres. Such substrates were synthesized using a robust and inert material, namely polydimethylsiloxane (PDMS). To increase its elasticity and the adhesion of the spheres to its surface a smaller than normal concentration (~0.1) of cross-linker was used during the curing step of the manufacture of the PDMS substrate.
The polystyrene spheres were deposited at the top of stretchable substrate according to the following steps. Firstly, a droplet of a mixture of spheres with different sizes was dried on a microscope slide. After that the slide with the spheres was pressed into the PDMS substrate to deposit the spheres at its surface. Next, the PDMS substrate was detached and stretched until it reached its maximum strain ~0.2. The spheres were then micromanipulated at top of the substrate under microscope control to form a bisphere with d~2-3 µm. After that the strain was released step by step using a translational stage to control d.
Since the distribution of strain in a centimeter sized substrate can be nonuniform, it is important to characterize the strain in the same area where the bisphere under study is assembled. For this purpose we used two additional microspheres as markers spanning the area of interest, see Fig. 1(b). The distance between the markers (L) was sufficiently small to ensure the uniformity of the strain, but it was large enough (~0.5 mm) to enable very simple and accurate optical calibration of the strain using imaging. As follows from the geometrical sketch in Fig. 1(b), the distance between the spheres is related to the strain by the formula:
where L — distance between two markers corresponding to the bisphere touching case, ΔL — increment of L due to stretching, D S(R) — diameters of two spheres (S and R) under study.
It is important to note that this simple technique provides a measurement of the separation d with nanometric accuracy, far beyond the limitations (~1 µm) of direct microscopic visualization of the gap sizes. This accuracy is determined by the possibility of precisely calibrating the strain (ΔL/L) by using distant markers with better than ~1% accuracy. This translates into ~30 nm accuracy in determining changes of intersphere separations d.
3. Experimental results
As illustrated in Fig. 1(c) the doped spheres were pumped at the center of the absorption band of the dye (467 nm) by a linearly polarized tunable OPO system with a repetition frequency of 20 Hz, and pulse duration of 20 ns. The illumination was provided by a slightly focused beam. Due to the fact that the incident light waves are not effectively coupled to WGMs the absorption was not influenced by the resonances in spheres. The excitation power was selected to be slightly above the lasing threshold for the WGMs peaks. The emission spectra display multiple TE- and TM-like WGMs peaks in 500–560 nm range. The individual cavities were imaged in the plane of spatial filter by using additional white light illumination through the transparent substrate using a Mitutoyo ×100 objective with a numerical aperture NA=0.5.
The typical example of results of characterization of such bispheres with controllable separation are presented in Fig. 2 for D s=5 µm and D r=8 µm. The spectra contain two well separated peaks around 518 nm and 526 nm associated with two different WGMs excited in the S sphere. The width of the resonances ~0.2 nm is limited by the resolution of the spectrometer. The weak coupling regime is evident from the fact that split components [15–17] do not appear in these spectra even for smallest separations between the cavities. In the touching sphere case this can be seen by comparing spectra in Fig. 2(a) and 2(b).
As the cavities become closer one can see a slight shift of spectral peaks to shorter wavelengths indicated as γ in Fig. 2. This shift is present in spectra of both cavities, and it has a similar magnitude for all peaks. The relative shift reaches its maximum value γ/λ~7×10-4 in the d=0 case. This shift is not related to strong coupling phenomena since it does not depend on the individual characteristics of the resonances involved such as detuning between WGM eigenstates in S and R sphere. The most likely explanation of this shift is connected with the changes of the average index experienced by the optical modes confined in spheres. The SEM images of the structure indicate that the spheres are slightly embedded (~0.5 µm) into elastomeric PDMS slab, as illustrated in the inset in Fig. 2. As a result, the modes confined in the polystyrene spheres with index n=1.59 are influenced by the closely located PDMS region with index n PDMS=1.45. The effective index experienced by these modes is slightly increased for this reason. In the course of releasing the tensile strain the spheres are pushed up by the shrinking material of the substrate that result in a short wavelength shift of peaks due to slightly smaller effective index experienced by the modes in spheres.
As a measure of the coupling efficiency (η) between S and R spheres we used the ratio of the amplitudes of scattering peaks induced in R sphere (IR), see Fig. 2(b), to the amplitudes of the corresponding peaks in the spectrum of isolated S sphere (IS), as shown in Fig. 2(i) (the distance d=3.2 µm is sufficiently long to consider the S sphere isolated). Since all spectra in Fig. 2 were obtained at practically identical conditions of collection of light, the changes in the intensity of scattering peaks can be attributed to variations of the coupling efficiency between S and R spheres. We additionally corrected coupling efficiency by a geometrical factor (A) related to a finite size of the hole in the spatial filter. In our experiments it corresponded to a size ~6 µm at the sample plane that means that FL from 5 µm S sphere was totally transmitted through the hole, whereas only a part (1/A=(6/8)2=0.56) of the image of the 8 µm R sphere was coupled to the spectrometer. Finally, a rough estimate of the coupling efficiency was obtained using the following formula: η=A×(IR/IS).
4. Discussion and summary
As illustrated in Fig. 3 the experimental dependence of η as a function of d demonstrates the following properties: (i) maximum efficiencies η=0.2-0.3 in touching case, (ii) somewhat reduced, but still remarkably high (η>0.1) efficiencies for inter-sphere separations up to d~0.3 µm, (iii) nearly exponential decay with ~0.2–0.3 µm attenuation length for longer separations 0.3 µm<d<1.0 µm, and (iv) small, almost constant level η~0.02 at 1.0 µm>d>2.3 µm.
In analogy to the case of atoms where the typical length of coherent interaction  corresponds to d~λ one can assume the possibility of coherent coupling between two cavities at small separations. The behavior of η at d<1 µm is found to be in a good qualitative agreement with the results of theoretical modeling  performed for smaller S (3 µm) and R (2.4 µm) spheres. It is important to stress that this modeling considers coupling between size-mismatched cavities in the situation where the WGM eigenstates in S and R cavities are strongly detuned. This is similar to the bispheres studied in the present work. In such cases the evanescent coupling has been shown to result from the Fano resonance between a discrete energy state (true WGM) in S sphere and a continuum of “quasi”-WGMs with noncircular shape and reduced Q factors which can be induced in R sphere. It should be noted however, that the exact amount of detuning is unknown in our experiments, since the R sphere was undoped making it difficult to observe its WGM eigenstates.
An interesting alternative mechanism  of coupling between spherical or cylindrical cavities is theoretically predicted in the case of their plane wave illumination due to the formation of localized “nanoscale photonic jet” at the shadow-side of spheres. In our experiments however, the excitation is provided by spherically symmetric WGM in the S sphere as evident from the spectroscopic data that shows that the photonic jet mechanism is not directly applicable in this situation.
A small and nearly constant level (η~0.02) of scattering detected from R sphere at larger separations between the cavities (1 µm>d>2.3 µm) cannot be explained by the model  of evanescent coupling described above. A likely explanation of this result is related to an illumination effect produced by the S sphere at the WGM frequencies due to scattering modes. A small fraction of such modes is reflected at the surface of R sphere into the objective, thus determining the level of background scattering peaks in our experiments. Due to its nonevanescent nature, this illumination effect depends on d weakly. However, as expected, the level of background scattering can be reduced with further increase of d (>2.5 µm).
In conclusion, we experimentally demonstrate high efficiencies (up to 0.2–0.3) of coupling between separated and size-mismatched cavities with strongly detuned WGM eigenstates. At small inter-cavity separations d<1 µm the coupling is explained by the mechanism  based on the Fano resonance between a discrete energy state (true WGM excited in the S cavity) and a continuum of quasi-WGM states which can be induced in the R cavity. The level of coupling efficiencies presented in this work leaves a room for further improvement which can be achieved by selecting more uniform cavities assembled as a long chain. However, the mechanisms of coupling between detuned cavities studied in this work should still play an important role in a variety of real physical systems with a degree of disorder.
The authors thank A. M. Kapitonov, M. S. Skolnick, A. M. Fox, D. M. Whittaker, and M. A. Fiddy for stimulating discussions. This work was supported by ARO under Grant No. W911NF-05-1-0529 and by NSF under Grant No. CCF-0513179 as well as, in part, by funds provided by The University of North Carolina at Charlotte. Shashanka Ashili and Charles Sykes were supported by DARPA Grant No. DAAD 19-03-1-0092. The authors are thankful to Duke Scientific Corp. for donating microspheres for the research presented in this work.
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