1. Introduction

Coupling of glass microspheres to optical waveguides through evanescent fields allows integration of compact high-Q resonators in optical circuits combined with a flexible choice of resonator material for lasing, switching, sensing or filtering devices. Coupling from a planar optical circuit results in robust construction, straightforward alignment of multiple and concatenated resonators, direct access to evanescent fields, high compatibility with optical fibre applications and low cost of fabrication [1, 2]. The coupling efficiency between waveguide and resonator is critically controlled by the separation between the microsphere and the waveguide [3, 4], allowing control of the power circulating in the sphere and transmitted through the coupled waveguide. Design of devices which employ coupling to microsphere resonators is considerably more complex than those using ring resonators, because microspheres support whispering gallery modes (WGMs) with a variety of azimuthal, polar and radial mode numbers, l, m and n, respectively. Each of these has its own coupling factor, Q-factor, effective radius, and resonant wavelength; although in the case of the perfect sphere the polar modes are degenerate. Successful exploitation of such microspheres in optical circuits requires a detailed understanding and control of the coupling from a waveguide mode to each whispering-gallery mode.

In this paper, evanescent coupling of light from channel waveguides into microsphere WGMs is studied at wavelengths between 1520 nm and 1610 nm. Experimental WGM resonance spectra are obtained for a small (∼15 μm radius) microsphere where two radial WGM orders are observed and a large (∼100 μm radius) microsphere where three radial WGM orders are observed. Experimental results are compared with theory by adapting the mode-coupling model of Little [5] to coupling to a planar channel waveguide to explain the relative magnitudes of the resonant peaks, and families of resonances are identified. Universal coupled microresonator theory [7] is used to fit the WGM spectra, assign WGM mode numbers to each resonance, and confirm the refractive indices and physical radii of the spheres.

2. Coupled resonator operation

2.1. Whispering gallery mode characteristics

The coupled resonator configuration studied is illustrated in Fig. 1, and consists of a sphere in proximity with a channel waveguide, coupled to it through evanescent fields. Operation of this system requires knowledge of the coupling factor κ which may be determined from the field distributions of the whispering gallery modes and the waveguide mode.

The characteristic equation resulting from matching tangential components of electric and magnetic fields at a sphere boundary [5] allows the wavelengths of the resonances and their corresponding modal field distributions to be found:

 

Fig. 1. Coupling configuration

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Fig. 2. WGM radial field distributions

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where:

α_s is the evanescent field decay constant away from the sphere in the radial direction, β_l is the modal propagation coefficient parallel to the surface of the sphere, R_o is the physical sphere radius, n_s is the refractive index of the sphere, n_o is the refractive index of the medium surrounding the sphere, and λ is the free-space wavelength.

l is the azimuthal mode number and is equal to the number of wavelengths taken to travel around the sphere for a particular resonance. For a particular value of l, there are many solutions of Eq. (1), due to the form of the spherical Bessel function j_l, and each one corresponds to a different radial mode number, n, with n being equal to the number of intensity maxima in the radial direction. The final, polar, mode number, m, describes the field variation in the polar direction, with the number of intensity maxima being equal to l - ∣m∣+1, so that the “fundamental” mode has l=m and n=1. Modes with the same values of l and n but different values of m are degenerate but, having different field distributions, will have different waveguide coupling factors and, hence, Q-factors.

Each solution of Eq. (1) yields the field distribution of a WGM and Fig. 2 shows example plots of the field distributions of the fundamental (l=83, n=1) and the second order (l=77, n=2) TM modes as a function of radial position, for a sphere that has a refractive index of 1.50 and a radius of 15 μm. These solutions exist at wavelengths of λ_83,1=1551 nm and λ_77,2=1549 nm, respectively. The field distributions shown in Fig. 2 are normalised so that equal power is carried in both whispering gallery modes.

2.2. Coupling to a channel waveguide

To determine the coupling factor between a channel waveguide mode and each WGM, a numerical overlap integral of the above WGM fields with the waveguide field calculated using the beam propagation method (Beamprop, RSoft) was carried out. The waveguide was modelled as a homogeneous channel of refractive index 1.52, of depth 2.3 μm and width 7.0 μm, embedded in a substrate of refractive index 1.50, yielding a theoretical modal depth and width, at λ=1550 nm, similar to the experimental waveguide reported in the next section. The microsphere was taken to have a radius of 15 μm and a refractive index of 1.50, and was spaced from the waveguide by a thin layer of refractive index 1.29, as shown in Fig. 3.

 

Fig. 3. WGM and waveguide fields

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Fig. 4. Coupling factor vs separation

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The coupling factor, κ , was calculated according to Eq. (6) and (7) [5]. κ² is the proportion of power coupled from the waveguide mode to a specific WGM per revolution.

where

E_s is the WGM field of the sphere and E_w(D_teflon) is the waveguide evanescent field as a function of Teflon thickness in x-y cross-sectional plane. E_s and E_w(D_teflon) are normalised such that ∬∣E_s∣² dxdy=1 and ∬∣E_w(D_teflon)∣² dxdy=1, respectively. δβ is the difference between the WGM and waveguide mode propagation coefficients. γ_wg is the waveguide mode decay constant away from the waveguide surface. K_o represents the coupling coefficient at the point of minimum sphere-guide separation. The integration is carried out over the lower half of the sphere in the x-y cross-sectional plane shown in Fig. 3. We emphasise that the model provided by Eqs. (6) and (7) only holds in the limit of small values of δβ, whereas for large values modifications along the line of Snyder and Ankiewicz [6] are required. However, we find that for the parameters considered here, e.g. δβ=0.54 μm^-1 for l=m=83, δβ=0.92 μm^-1 for l =m=77, the simple model gives accurate agreement with the experimental data.

Figure 4 shows the coupling factor, κ, as a function of the waveguide-sphere separation for the TM polarisation, for four WGM’s resonant at wavelengths close to 1550 nm. The strongest coupling is shown for the fundamental radial mode resonant at 1551 nm (l,m,n=83,83, 1), while weaker coupling is observed for the second-order (l,m,n=77,77, 2) and third-order modes (l,m,n= 72,72, 3) resonant at 1549 nm and 1550 nm, respectively. The mode of order (l,m,n=83,81, 1) is the third-order polar mode which is degenerate with the (83,83,1) fundamental mode. It can be seen that κ is greater for n=1 than n=2, despite the intensity at the sphere boundary being lower for n=1 than for n=2, as the integral of the field overlap is greater when n=1.

2.3. Circulating power spectra and waveguide coupling

The transmission characteristics of the microsphere-coupled waveguide, and the power circulating in the microsphere, may be described using universal coupled microresonator theory [7]. In Fig. 1, the field amplitudes at the input and the output of the waveguide are denoted a ₁ and b ₁, respectively, while the field amplitudes in the sphere immediately before and after the coupling region are a ₂ and b ₂. The coupling factor, κ, calculated using Eq. (6), and transmission coefficient, t, are related by ∣κ∣²=1-∣t∣². α is the circulation loss factor, $\alpha =\frac{{4\pi }^2{n}_{\mathrm{eff}}{R}_o}{\lambda },$, which is unity for a sphere with zero loss, and φ is the circulation phase shift, given by:

where n_eff corresponds the effective refractive index of the mode that propagates along a circular path within the cavity. The power circulating in the sphere is related to the input power in the waveguide by [7]:

where φ_t is a phase offset due to coupling to the waveguide.

Typical curves of circulating power versus wavelength for a sphere of refractive index 1.5, radius 15 μm, circulation loss factor of 0.9946 (equivalent to a propagation loss of 5 dB/cm) and coupling factors ranging from 0.1 to 0.9 are given in Fig. 5. In the case of a sphere with known loss, the effective optical path length (2πn_effR_o) and the coupling factor, κ, can be readily extracted from the experimental data for scattering using Eq. (9).

The power circulating in the sphere in each mode at its resonance is dependent upon the coupling factor and the circulation loss factor,α, as shown in Fig. 6 for α = 0.9946, where it is maximised at κ ∼ 0.1, which corresponds to κ_c = [1-∣α∣²]^1/2 from Eq. (9). This corresponds to critical coupling where the lost power per round trip is equal to the coupled power. For κ < κ_c, the microsphere is undercoupled and the circulating power increases with coupling factor. The opposite behaviour is observed in the overcoupled case where κ >κ_c.

Figures 4 and 6 show that the microsphere circulating power is a strong function of both the sphere loss and its separation from the waveguide and that, for maximum circulating power, there is an optimum separation dependent upon the sphere loss corresponding to critical coupling. Optimisation of the circulating power is crucial for devices which exploit microsphere nonlinearity or lasing.

3. Coupled resonator measurement procedure

Experiments were carried out to determine the characteristics of coupling from an optical waveguide to a glass microsphere, including Q-factor, estimation of sphere radii, and mode assignment. Ion-exchange was carried out in BK7 glass from an AgNO₃/KNO₃/NaNO₃ melt of composition 0.5:49.75:49.75 mol% at 350°C through mask openings of 4 μm width, for 4 hours, to yield monomode waveguides at wavelengths near 1550 nm, and the ends were polished to allow fibre butt coupling. A Teflon AF2400 (DuPont) film (n_teflon ∼ 1.29) of thickness 400 nm was deposited on part of the waveguide to allow simple separation of the sphere from the waveguide. Laser light, tunable from 1440-1640 nm (Agilent 81600B) was focused into a polarisation maintaining fibre and coupled into the waveguide (in the TM polarisation) by butt-coupling, as shown in Fig. 7.

 

Fig. 5. Circulating power vs wavelength

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Fig. 6. Circulating power vs coupling factor

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Two types of microspheres were studied. First, commercially Nd³⁺-doped BK7 spheres of 15±1.5 μm in radius (Mo-sci) and, second, Nd³⁺-doped BK7 nominally 100 μm in radius fabricated in our laboratory. Neodymium is incorporated into the spheres to allow subsequent lasing studies [8, 9, 10] and this will influence the refractive index and losses of the spheres. In each case, the microsphere resonator under test was positioned on the Teflon film directly above the waveguide. A microscope with CCD camera was used to image the microsphere and an InGaAs detector was used to measure the light scattered into the microscope from the microsphere. The position of the sphere was optimised by translating it across the waveguide and maximising the scattered power detected on resonance. In this experiment, the microsphere Q was limited by surface roughness and therefore the power scattered from the microspherewas presumed proportional to the circulating power for a given mode. Both the power scattered and the power emerging from the waveguide were recorded as a function of laser wavelength.

 

Fig. 7. Measurement apparatus

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4. Coupled resonator results and discussion

4.1. 15 μm radius sphere

Figure 8 shows the scattered power collected through the microscope by the InGaAs detector as a function of wavelength, with fitted theoretical curves for circulating power, for the case of the 15 μm radius BK7 spheres.

 

Fig. 8. Measured scattered power for 15 μm BK7 microspheres

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The experimental data show high amplitude “strong” resonances with a free spectral range (FSR) of 17.380±0.005 nm at the centre wavelength and a family of “weak” resonances with an FSR of 18.138±0.002 nm at the centre wavelength. The free spectral range (FSR) is related to the effective index of a mode by:

The theoretical curves in Fig. 8 are plots of Eq. (9) fitted to the experimental data separately for the n=1 and n=2 modes, as follows. The coupling factors are calculated using Eq. (6) and the FSR is used with Eq. (8) to determine the circulation phase shift, φ. The circulation loss factor, α, is then adjusted so that the widths of the resonant lobes in the theoretical and experimental data are well matched. The experimental data has been normalised so that the magnitudes of the resonances of the fundamental modes match the theoretical results. The main peaks are found to exhibit a Q-factor of 4000 and a circulation loss factor of α= 0.93, using the calculated coupling factor of 0.044 obtained from Eq. (6) in Fig. 4. The value of Q is found from the fitted curves using $Q=\frac{\lambda }{{\mathrm{\delta \lambda }}_{\mathit{FWHM}}}$, where δλ_FWHM is the width of the resonance lobe at full-width half-maximum power. The theoretical plot for the second-order (n=2) mode, using the same circulation loss factor and a calculated coupling factor of 0.014, shows resonance magnitudes relative to the fundamental (n=1) mode which are well matched with the experimental data. The same circulation loss factor is used for all modes as the fit of mode width for the higher-order modes is insufficiently accurate to justify using another value. The fluctuating background is probably due to non-resonant scattering of light from the laser source.

The mode numbers n=1 and n=2 are assigned to the “strong” and “weak” families of peaks, respectively, with the following justification. In an experiment limited by surface roughness, the measured scattered power will depend upon the circulating power in the mode and upon the scattering efficiency from that mode which, in turn, depends upon the surface intensity of that mode for a given modal power. The solutions to the characteristic equation (Eq. (1)) can be used to calculate the field distribution in the sphere for a given modal power, as shown in Fig. 2, and it is found that the surface intensity for the (l,n)=(83,1) mode is 60% of that for the (l,n)=(77,2) mode, so that the fundamental mode would be expected to scatter less, for equal modal power. However, for the expected values of κ , which are 0.044 for the fundamental and 0.014 for the n=2 mode, it can be concluded from Fig. 6 that for the low circulation loss factors observed here (α ࣘ0.93), the circulating power increases rapidly with κ , and so will be much higher in the fundamental mode than in the second-order mode. This will result in the highest scattered power for the fundamental mode despite the lower normalised surface intensity for this mode, justifying the assumption that the mode exhibiting the strongest scattering is the fundamental mode.We also attempted to match the weaker family of peaks to TE excitation with no success, confirming that this family was not due to spurious TE excitation.

Precise assignment of mode numbers to the experimental data and extraction of the physical radius (R_o) and refractive index (n_s) of the sphere from these data requires matching theoretically predicted resonant wavelengths with all the experimentally measured resonances, using the characteristic equation (Eq. (1)). The refractive index of the BK7 spheres was estimated to be 1.5004 at λ =1570 nm using the Schott datasheets and the Sellmeier equation [11]. Starting with this refractive index and the nominal sphere radius of 15 μm, the resonant wavelengths corresponding to zeroes of Eq. (1) for specific mode numbers l, were found for radial mode numbers n=1 and n=2 at wavelengths between 1520 nm and 1610 nm. The sphere radius and index were then adjusted until the best correspondence between experimental and theoretical resonant wavelengths was achieved.

Assigning the major peaks to fundamental WGMs with n = 1 yields azimuthal WGM numbers ranging from l=m=87, for the resonance with the shortest wavelength shown in Fig. 8, to l=m=;3 for the longest wavelength resonance. Similarly, assigning the weaker peaks to the higher order radial mode, n=2, yields azimuthal mode numbers ranging from 80 to 76. The values of refractive index and physical radius of the sphere that provide the best match of all wavelengths for the two families of modes are n_s=1.5004 and R_o=15.387±0.007 μm. The full set of measured and theoretically predicted resonant wavelengths and assigned mode numbers, l and n, are shown in Table 1, with the deviation between the experimental and theoretically fitted values, δλres. The estimated tolerance on the deduced radius of the sphere reflects the range of modelled radii required to fit each resonance, in turn, exactly.

Table 1. Experimental and theoretical resonant wavelengths and assigned mode numbers l,n for the 15 μm sphere

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Good agreement between experimental and theoretical resonant wavelengths is achieved, so that mode numbers and sphere radius and index may be deduced with confidence, and the radius is found to be within the manufacturer’s tolerance of 15±1.5 μm. The relative magnitudes of the resonances for the fundamental and second order families of radial modes give confidence in the theoretical estimation of coupling factors, and the spectral widths of the resonances allow deduction of the circulation loss factor for the fundamental mode. The effective refractive index n_eff for each mode can be obtained by using Eq. (10) using the value of FSR found from the experimental data. The n_eff of the fundamental and the second order radial modes around a wavelength of 1550 nm are found to be 1.441 and 1.407, respectively. This is in line with the fact that the (l,n)=(83,1) mode is more confined inside the sphere than the (l,n)=(77,2) mode, as shown in Fig. 2. The loss factor of 0.93 is rather low for spheres of this small diameter, representing a round-trip loss of 0.63 dB and leading to a Q-factor of only 4000.

4.2. 100 μm radius sphere

In order to study the excitation of higher-order radial modes, larger spheres were studied as, for the present geometry, the larger radius leads to a larger circulating power in each mode due to the increased coupling factor, as shown in Eq. (6), which is primarily due to the longer effective coupling region.

New spheres were fabricated by crushing and carefully sieving Nd-dopedBK7 glass to obtain particles of radius of order 100 μm, and then passing this powder through a vertical furnace at 1250°C in an inert atmosphere [12]. While passing through the hot zone, spherical particles were formed by surface tension and these solidified as they entered the cooler zone. Spheres made by this method had radii of order 100 μm and have, to date, shown Q factors as high as 6.2×104, due largely to improved surface quality and resultant reduced surface scattering.

 

Fig. 9. Resonance spectrum for the 100 μm microsphere

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Figure 9 shows the experimental results obtained for scattered WGM power vs wavelength for an in-house made sphere with a radius of 100≤1.5 μm measured using an optical microscope, obtained in the same way as the spectra in Fig. 8. The free spectral range of the strongest (n=1) family of peaks is found to be 2.655±0.003 nm at the centre wavelength, and these modes showed Q-factors of 2.3×104. Two other families of peaks are observed, and the FSR of the second and third families are 2.671±0.001 nm and 2.70±0.01 nm at centre wavelengths, respectively. The same fitting procedures were carried out for the three families of modes (n=1,2, 3) in this data as for the two families observed for the 15 μm sphere. The coupling factors calculated for a sphere of radius 100 μm separated from the waveguide by a 400 nm Teflon film are calculated, using Eq. (6), to be 0.0935, 0.0473, and 0.0274, for the n=1,2,3 modes, respectively, so that, as before, the n=1 mode is expected to show the strongest peaks, and the n=2 and n=3 modes to show correspondingly lower scattered power. Fitting Eq. (9) to the fundamental family of resonances results in a circulation loss factor of 0.90, and the relative magnitudes of the theoretically calculated circulating powers for these values of κ and α are in good agreement with the experimental results. A circulation loss factor of 0.90 corresponds to a round-trip loss of 0.87 dB. While this is higher than for the smaller spheres, the Q is greater because the larger sphere has more stored energy at resonance and $Q\approx \frac{{2\pi }^2{n}_{\mathit{eff}}{R}_o}{\lambda \left(1-\alpha \right)}$[13.

The experimentally observed resonant wavelengths were fitted to the characteristic equation and mode numbers l and n assigned to each resonant wavelength, as before, by adjusting the radius, R_o, and the sphere index. Table 2 shows the full set of experimental and theoretical resonant wavelengths for the fundamental, second and third order radial modes (n=1,2, 3), for the best-fit values of R_o=99.31±0.03 μm and n_s=1.5006. The discrepancies between the theoretical and the experimental resonant wavelengths are less than 0.5 nm for all modes. Sphere mode numbers (l,n) of (590,1) to (585,1) are assigned to the major peaks, corresponding to fundamental radial WGMs. The sphere mode numbers of (578,2) to (573,2) and (568,3) to (563,3) are assigned to the second and third order radial WGMs. The corresponding values of n_eff at a wavelength of 1550 nm for the fundamental, second, and third order radial modes are 1.457, 1.450 and 1.435, respectively. This is again in agreement with the fact that higher radial number modes extend further into the surrounding air.

Table 2. Experimental and theoretical resonant wavelengths and assigned mode numbers l,n for the 100 μm sphere

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The Q factors of the coupled 100 μmmicrospheres are significantly improved compared with the as-supplied spheres, due to improved surface quality. The circulation loss factor,α, is lower for the larger spheres, however, because of the longer WGM path length. The individual mode numbers have been identified with confidence, the physical radius and index of the spheres have been confirmed, and waveguide coupling to each mode is in agreement with theory. The attempts at matching the peaks with TE excitation have failed, validating that they were not due to TE excitation.

5. Conclusions

A detailed experimental and theoretical study of the coupling of ion-exchanged optical waveguides with a low-index isolation layer to whispering gallery modes of Nd:BK7 spheres of different sizes has been carried out. Fitting the experimentally observed resonance spectra to universal coupled microresonator theory has allowed extraction of sphere loss factors, using coupled mode theory to estimate sphere/waveguide coupling factors. Assignment of mode numbers l and n to each of the observed resonances has been achieved by comparison with the characteristic equation for microsphere resonators. Excellent agreement has been achieved between theory and experiment, in terms of mode position, spacing, width and amplitude, and the precise fitting of all modes has allowed extraction of the physical radius and refractive index of the spheres as free parameters in the fitting procedure.

Commercial Nd:BK7 spheres were found to have high losses due to significant surface roughness. Higher-quality Nd:BK7 spheres were prepared by crushing and sieving glass to realise a refined powder and heating while being dropped through a vertical tube furnace to allow surface tension to form microspheres before reaching a cool zone. Experimental results show a significantly improved Q-factor. Further improvements in sphere quality by optimisation of the fabrication process, and construction of more complex waveguide structures, are expected to lead to novel devices including integrated microsphere laser circuits. These results are expected to be extremely useful in the design of efficient, fully integrated microsphere lasers.