The inelastic emission spectrum of a single fluorescent microsphere (bead) exhibits resonances arising from whispering gallery modes. Two beads in close proximity form a coupled bisphere. Coherent coupling arises from each bead’s evanescent field and leads to resonance splitting. Here we collect emission spectra of two coupled beads, with nearly identical diameters, as spacing between beads is varied. Using these size-matched beads allows us to probe resonance splitting under strong coupling conditions.
© 2010 OSA
Light waves propagating circumferentially around the inside surface of a dielectric sphere can be efficiently confined by total internal reflection. Then a resonant standing wave arises inside the sphere at wavelengths where the light returns in phase. The natural frequencies of oscillation at which these resonances occur are described as whispering gallery modes, and the resonances themselves are referred to as morphology dependent resonances (MDRs).
That resonances due to whispering gallery modes arise in the emission of fluorescent spheres has been known for some time. A half-century ago, lasing from 1 mm spheres of fluorescent CaF2: Sm++ was interpreted as stimulated emission from whispering gallery modes . The MDRs arising in a sphere’s fluorescence emission correspond to isolated singularities in the coefficients of a multipole expansion of the sphere’s fluorescence field [2,3]. The same coefficients appear in the multipole expansion of plane wave scattering from a sphere . The whispering gallery modes and corresponding MDRs can be separated according to their transverse magnetic (TMsn) or transverse electric (TEsn) nature, where the mode number n indexes the multipole coefficient and indicates the number of wavelengths around the sphere’s circumference, and where the order number s identifies a singularity in a multipole coefficient and corresponds to the number of maxima in the radial direction. The modes are also denoted by an axial mode number m that determines the axial orientations of the circular path, but for an isolated sphere the MDRs are independent of m, so that each MDR has (2n + 1)-fold degeneracy. With knowledge of sphere diameter and refractive indices of the sphere and surrounding media, the multipole coefficients can be used to determine peak wavelengths and natural widths of MDRs.
Ashkin and Dziedzic first observed optical MDRs in radiation pressure spectra of optically-levitated oil drops . Then Benner et. al. identified optical MDRs in the fluorescence emission of dye-doped, 10 μm, polystyrene spheres , where they appear as sharp peaks in the inelastic emission spectra of the spheres. They also demonstrated that wavelengths at which MDR peaks occur depend strongly on the sphere’s diameter.
Since then, polystyrene microspheres (which we'll refer to as beads) volume-doped or surface-coated with fluorescent dye have become a common experimental tool for studying optical MDRs. Kuwata-Gonokami and Takeda examined inelastic emission spectra of beads with different diameters and different dye-penetration depths . They found that beads with a surface coating of dye allowed s = 1 order resonances while suppressing higher order resonances that are present in dye-doped beads. This order suppression occurs because field amplitudes of s = 1 resonances are largest near a sphere’s surface while maximum field amplitudes of higher-order resonances are located further inside the sphere . Suppression of higher-order MDRs for dye-coated beads can also be seen in spectra collected for other studies [8,9].
The mode structure of light circling around a dielectric sphere is reminiscent of quantum mechanical states of an electron in a hydrogen atom, giving rise to a photonic atom description [10,11]. Then bringing together two spheres creates a bisphere system that can be described as a photonic molecule [12,13]. In the case of a hydrogen molecule, electron-electron interaction removes the degeneracy of atomic hydrogen states, giving rise to energy splitting of molecular states. Similarly for bispheres, coherent coupling of the evanescent fields extending beyond each spheres’ surface removes the degeneracy of axial modes contributing to MDRs. Since the evanescent field outside each sphere decays exponentially with increasing distance [11,14], coherent coupling of axial modes and the resulting mode splitting of MDRs should be very sensitive to the spacing between spheres.
Mukaiyama et. al. studied coupled bispheres composed of two dye-doped beads in contact with each other , where they examined the dependence of mode splitting as relative size of beads was varied. Möller et. al. also examined coupled bispheres in contact, composed of dye-coated beads , where they reported on the strength of coherent coupling at various locations along the bisphere. They also mention that no coherent coupling is observed when separation distance between beads equals their diameter.
Ilchenko et. al. performed the first study of coupled bispheres, where they examined the dependence of mode splitting upon separation distance between spheres . The investigation highlights the case of TE mode coupling, since field amplitudes of TE modes extend further beyond the surface of a sphere than those of TM modes. They separate coherent coupling of bispheres into two ranges, which we can describe as weak and strong coupling. Weak coupling occurs when the MDR peaks in the two spheres’ spectra are not well-aligned, i.e., when the wavelength spacing between their peaks is large. Under this condition the wavelength splitting of modes arises from two separate mechanisms. Then, of course, strong coupling occurs when MDRs in the two spheres’ spectra are well-aligned; under this condition the wavelength splitting Δλ of a mode with peak wavelength λ is dominated by a single mechanism:Eq. (1) as the overlap . For the specific case where the spheres have nearly identical diameters, so that TE modes sharing the same mode numbers are well-aligned, the overlap takes the form14]. The r* in Eq. (2) is the field decay parameter. For spheres with refractive index n p in a surrounding medium of refractive index n w, the field decay parameter takes the form
Experimentally, Ilchenko et. al. investigated coherent coupling of bispheres composed of silicon spheres as separation distance between spheres was varied . MDRs appear in the absorption spectrum of laser light coupled to one of the spheres via a prism. The size of the silicon spheres (70-200 μm diameter) leads to a large density of MDRs at optical wavelengths, so that their measurements were conducted on unidentified modes with random alignment, e.g., weak coupling conditions. Their measurements matched well with predictions assuming the weak coupling condition of their theory.
Here, we present inelastic emission spectra of bispheres formed from a pair of dye-doped beads matched in size, i.e., beads with nearly the same average diameter, as the spacing between them is varied. By using beads with nearly the same average diameter, each MDR peak appearing in one bead’s spectrum aligns well with a resonance peak in the other’s spectrum that shares exactly the same polarization and mode numbers. This alignment allows us to probe the strong coupling condition of the theory by Ilchenko et. al. .
2. Experimental setup
The positioning of two beads in close proximity to each other is accomplished by holding the beads in two independent optical traps of an optical tweezers; a schematic of the tweezers is shown in Fig. 1 . This tweezers employing two single-beam gradient traps is based on the “open microscope” design of Bechhoefer and Wilson . The central element is a Plan Apochromat 63X 1.4 NA oil immersion microscope objective lens (Zeiss 440766-9901-000). An 830 nm, 150 mW, microlensed diode laser (Power Technology PPM830-150B) provides the incident beam that is split to form the two optical traps. The polystyrene beads used are transparent to this laser’s infrared wavelength, so that they can be trapped without absorptive heating. A telescope expands the beam to overfill the back aperture of the objective lens. Then an optical assembly, composed of two polarized beamsplitter cubes and two mirrors, separates the incident beam into two independent, equal-power beams directed along parallel paths. A dichroic mirror (Chroma Z488/830RPC) inserts the two beams into the objective. The objective converges the beams, creating the two focal spots corresponding to the optical traps’ equilibrium positions. The location of each trap in the objective’s focal plane is determined by the corresponding mirror in the beam-separation assembly, which is held in a stepper-motorized kinematic mount (Thorlabs KS1-ZST). The smallest step used to tilt a mirror corresponds to moving an optical trap less than one-fifth the distance imaged by one camera pixel.
A 473 nm, 20 mW DPSS laser (Laserglow LRS-0473-KFM-00030-05), with output power reduced to 0.02 mW, provides the beam used to excite fluorescence in trapped beads. A telescope expands the beam to overfill the back aperture of the objective lens, and then a dichroic mirror (Chroma Z488RDC-XT) inserts the excitation beam along a path parallel to those of the two infrared beams. Then, as with the infrared beams, the first dichroic mirror inserts the excitation beam into the objective. The objective converges the excitation beam, creating a focal spot in the same plane as the two infrared focal spots. The excitation beam’s focusing geometry is identical to that of the two infrared beams. So without power reduction, it can also be used as an optical trap, which allows us to locate the excitation focal spot.
The objective lens collimates light originating from a point on its focal plane, and then a subsequent tube lens converges this light to create a real image. Filters inserted after the tube lens eliminate reflected light from the excitation and trapping lasers. Then a 30% reflective mirror separates the converging light, creating two image planes.
A CCD camera (JAI-Pulnix TMC-6740GE) is located at the reflected image plane. The 640x480 square-pixel CCD is aligned to capture the central region of the real image produced by the tube lens. Each photograph presented here was collected using a 5 ms exposure time.
A 50 μm diameter fiber optic connected to a spectrometer (Ocean Optics HR4000) is located at the transmitted image plane. The spectrometer records inelastic emission spectra from the fluorescing beads, covering a spectral range 500 – 600 nm. The range of interest is covered in 0.035 nm steps with 0.10 nm resolution. Each spectrum presented here is composed from 20 scans, each collected over a 500 ms integration time. Comparing the width of the spectrometer’s fiber optic to the camera’s pixel width, the fiber optic’s diameter corresponds to 7 pixels. The fiber optic is mounted to an x-y-z translation stage that allows it to collect light from any region of the real image.
The sample chamber placed in front of the objective is composed of a thin coverslip and microscope slide, separated by a U-shaped silicone bead to form a cup. The cup is filled with a colloidal solution of beads in distilled water and mounted on an x-y-z translation stage, so that the oil-coated coverslip faces the objective. Transmitted-light photographs of 2 μm beads trapped by each laser are used to locate the focal spots of all lasers in the image plane. Repeated comparisons between the readings from a stepper-motor used to translate the sample chamber and the pixel positions of a bead stuck on the chamber’s coverslip in transmitted light photographs allows us to determine a value of 0.114 μm for the length imaged by a single camera pixel.
The beads used here are custom 10 μm diameter fluorescent green ring-stained polystyrene microspheres (Invitrogen Custom Services FocalCheck microspheres, C29351). The beads are used as received from the manufacturer. These beads’ proprietary volume-penetrating dye has a free-space emission extending from 510 – 610 nm with a peak at about 530 nm. In the custom order, the depth of dye penetration was minimized so as to suppress resonances with order . A primary benefit of this dye is its stability against photobleaching. In our setup, well over a hundred spectra can be collected from the same bead before photobleaching renders emission too low to use. However surface roughness, which degrades resonance widths , varies from one bead to the next. Only beads with observed resonance full-widths at half-maximum (FWHM) less than 0.40 nm, similar to those seen in the spectra of Schiro and Kwok , are selected for study; this condition is met by about 1% of beads examined.
Another experimental difficulty encountered is a small repulsive force between a pair of trapped beads. This repulsion is present whether or not the beads are fluorescing. The manufacturer confirms that the beads may have a residual anionic charge that, for example, prevents the beads from aggregating. So an electrostatic force appears to be the primary component of the repulsion. As one trapped bead is moved close to the other, the repulsive force causes the bead centers to shift away from the optical traps in order to provide a balancing restoring force. As a result, the repulsive force between beads prevents us from using optical trap locations to determine separation distances between beads.
We have noticed that moving one bead toward the other along a path parallel to a trapping laser’s polarization axis reduces the repulsion. This directional dependence suggests the presence of a smaller force between dipoles induced in the beads by the trapping lasers.
Coherent coupling of MDR evanescent fields can also induce a force between the beads [16,18], which may mitigate or exacerbate the electrostatic repulsion. Bead reaction to this force would appear as a change in separation distance when the beads are fluorescing, compared to when they are not fluorescing. However, comparing photographs of fluorescing beads and non-fluorescing beads backlit with white light, we observe no such change.
3. Measurements and results
Collecting an inelastic emission spectrum from a single isolated bead involves positioning it relative to the excitation laser beam’s focal spot and positioning the spectrometer’s fiber optic. Since the polystyrene bead can absorb some fraction of the excitation light, to avoid heating it during spectral collection, the bead is positioned so that the closest point on its surface is one micrometer from the excitation beam’s focal spot. With this spacing, the bead’s resonance emission is acceptable while the bead itself suffers no heating effects, even after a hundred spectral collections.
Figure 2 shows photographs of fluorescence emission from isolated beads. In these photographs, the emission appears as concentric diffraction rings with two ubiquitous bright spots located along the zero-order ring, where the bright spot on the left edge is closest to the excitation laser’s focal spot. These bright spots are the “poles” where the longitudinal paths of all axial modes cross. In each photograph, the diameter of a bead’s zero-order diffraction ring measured along the vertical is 88 pixels; multiplying this value by the 0.114 μm length of a single pixel gives a result very close to the nominal bead diameter.
Figure 3 shows the emission spectrum collected from each isolated bead appearing in Fig. 2. These spectra were collected by centering the spectrometer’s fiber optic on the bright spot along the right edge of a bead, opposite the excitation beam’s focal spot. In these spectra, the MDRs appear as narrow peaks superimposed upon a free-space emission background.
We can see that diameters of beads in Fig. 2 are closely matched by comparing MDR peaks appearing in their emission spectra in Fig. 3; if peaks appearing in one isolated sphere’s spectrum align at the same wavelengths as those in the other’s, then the spheres have the same diameter. The peaks in our two spectra are not perfectly aligned; resonances peaks in one spectrum are offset from corresponding peaks in the other spectrum by 0.07 nm. This offset is five times smaller than resonance FWHMs. To identify the resonance modes responsible for each peak in the spectra of Fig. 3, we calculated coefficients of the multipole expansion by Chew , using methods outlined by Zijlstra et. al. . Using refractive indices n p = 1.59 for the polystyrene beads and n w = 1.33 for the surrounding water leaves the bead diameter as the only free parameter in these calculations. The bead diameter was adjusted from an initial value of 10 μm until s = 1 resonance peaks in the calculated spectrum aligned with those in the measured spectra.
Ilchenko et. al. define the strong coupling condition for aligned resonances through the inequality α >> ξ 2 + Q −2 , where Q is the resonances’ quality factor, ξ is the initial offset of the resonance peaks, and α is the overlap defined in Eq. (2). Since the theory focuses on TE modes, we will use the TE1 85 resonance peak at 547.4 nm as a test case. Dividing resonance peak position by its FWHM gives the quality factor. Then, the TE1 85 peak appearing in spectrum (a) of Fig. 3 has an inverse quality factor Q −1 ≈6.4 x 10−4. Next, the initial offset of the resonance peaks is defined as the difference between peak positions in the two spectra divided by one of the peak positions. So the TE1 85 peaks spectra (a) and (b) of Fig. 3 have an initial offset ξ ≈1.3 x 10−4. Noting the resonances’ mode number n = 85 and recalling the estimate for ζα, we use Eq. (2) to calculate the amplitude of overlap α ≈1.8 x 10−3. Then for the TE1 85 peaks, the quantity ξ 2 + Q −2 is four thousand times smaller than the amplitude of the overlap α. So the strong coupling condition is met for any separation distance between beads smaller than, for example, the field decay parameter.
Spectral collection of fluorescence emission from a bisphere composed of the beads in Fig. 2 begins with the single-bead geometry described above, where the excitation focal spot is located to the left of a bead. This first bead remains stationary throughout the measurements with the spectrometer’s fiber optic centered on the bright spot along the right edge of the bead. The second bead is positioned some distance from the first on its right side. Then the second bead is moved toward the first in discrete steps, until it comes into contact at the first bead’s right-side bright spot. This contact point ensures maximum coherent coupling between the beads’ axial modes [13,15]. At each step, an emission spectrum is collected, along with thirty photographs of the fluorescing beads.
Figure 4 shows example photographs of fluorescence emission from the matched beads when they are in contact and when they are not in contact. We use the photographs to determine the separation distance between beads at each spectral collection step. In each photograph, we ascertain the pixel spacing between opposing edges of each bead’s first-order diffraction ring (the zero-order diffraction rings are not used because the bright spots on them blur their edges). Then we determine the average pixel spacing from all photographs captured during the collection step. We identify when the beads come into contact as the spectral collection step where this pixel spacing is minimum. Then the difference between this minimum pixel spacing and the average pixel spacing determined at other spectral collection steps determines the bead separation distances at those steps.
Figure 5 . shows emission spectra from the matched beads collected at different separation distances. Spectrum (a) in Fig. 5 was collected when the two beads are in contact. In this spectrum, the coupled axial modes result in pairs of TM peak intensities split by 0.81 nm and pairs of TE peak intensities split by 0.70 nm. We can compare the TE resonance splitting with an estimate from the strong coupling theory using the TE1 85 resonance. The separation distance between beads is x = 0, so that, as previously shown, the overlap α ≈1.8 x 10−3. Then substituting that overlap and the peak wavelength of the resonance into Eq. (1) gives an estimated splitting, Δλ ≈0.49 nm. Thus the estimated resonance splitting is close to the measured value, especially considering our assumption that ζα is constant.
Spectra (b) – (e) in Fig. 5 were collected as one bead moved away from the other. As expected, mode splitting decreases with increased spacing between beads. Figure 6 shows average TM mode splitting and average TE mode splitting plotted against separation distance. Fitting an exponential decay curve to each data set, we determine field decay parameters r* = 0.69 nm for TM resonances and r* = 0.70 nm for TE resonances. We can compare the TE resonances’ field decay parameter with an estimate from the strong coupling theory. Substituting numeric values for refractive indices of the polystyrene beads, n p, and the surrounding water, n w, and peak wavelength of the TE1 85 mode, λ, into Eq. (3) gives an estimate of the field decay parameter, r* = 0.10 nm.
Thus the experimental field decay parameter is seven times the estimated value. This discrepancy between experimental and estimated values may represent a breakdown in an approximation used in the theory ; namely, the replacement of Hankel functions, which describe the radial behavior of MDR field amplitudes outside the sphere , with exponential-decay functions that give rise to the field decay parameter. This approximation is valid for MDRs with large mode numbers, but optical-resonance mode numbers in our beads are an order of magnitude smaller than those of the silicon spheres used by Ilchenko et. al. .
The spectra (a) – (e) in Fig. 5 also display another curious feature; as soon as the beads begin to separate, the asymmetry in coupled-mode intensity begins to reverse. The spectrum (a) collected when the two beads are in contact, displays an asymmetry in mode splitting intensity that agrees with observations by Mukaiyama et. al. ; for each MDR, peak intensity of shorter-wavelength coupled modes is larger than that of longer-wavelength coupled modes. But in spectrum (b), where the beads first lose contact, the peak intensity of shorter-wavelength coupled modes is about equal to that of longer-wavelength coupled modes. Then as distance between beads increases, in spectra (c) - (e) the peak intensity of shorter-wavelength coupled modes is less than that of longer-wavelength coupled modes.
Finally, spectrum (f) in Fig. 5 shows no observable mode splitting when the beads are separated by 1 μm. However the influence of one bead upon the other is still seen as a reduction in MDR intensity, compared to that appearing in the spectra of isolated spheres in Fig. 2.
We have also examined the case where two beads differ in diameter such that there is no wavelength overlap between widths of MDRs appearing in their spectra. In this case, resonances appearing in the spectrum of one bead remain constant in shape and intensity even as the second bead is brought into contact with it. At all separation distances between beads, the resonance emission of one bead is unperturbed by the presence of the other bead, so that any coherent coupling between beads is too small for us to observe.
We identified inelastic emission spectra of two isolated fluorescent microspheres where identical morphology-dependent resonances modes appearing in the spectra are well-aligned. We used these spheres to create a coupled bisphere, in order to probe resonance splitting under strong coupling conditions. When the two spheres were in contact, the resonance splitting observed in the emission spectrum agreed with the theoretical estimate. However the field decay parameter, determined from resonance splitting in emission spectra collected as separation distance between spheres was varied, was seven times larger than the theoretical estimate. Also, in the spectrum collected when the spheres were in contact, shorter-wavelength coupled modes exhibited larger intensity, verifying previous experimental results. However, as spacing between the spheres increased, intensity shifted to longer-wavelength coupled modes.
This work was supported by Cottrell College Science Award #6845, a grant from the Research Corporation. Funding to build the experimental setup was provided by University of North Alabama research grants.
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