## Whispering gallery mode spectra of channel waveguide coupled microspheres

Optics Express, Vol. 16, Issue 15, pp. 11066-11076 (2008)

http://dx.doi.org/10.1364/OE.16.011066

Acrobat PDF (374 KB)

### Abstract

Evanescent coupling of light from glass channel waveguides into the whispering gallery modes of glass microspheres of radius 15 μm and 100 μm is studied experimentally at wavelengths near 1550 nm. Fitting the positions, widths and heights of resonances in the experimental spectra to the characteristic equation for microsphere modes and to universal coupled microresonator theory, we establish sphere radius and index, identify mode numbers, and determine losses. The results provide detailed information for the design of optical devices incorporating microsphere resonators in planar lightwave circuits.

© 2008 Optical Society of America

## 1. Introduction

1. J. P. Laine, B. E. Little, D. R. Lim, H. C. Tapalian, L. C. Kimerling, and H. A. Haus, “Microsphere resonator mode characterization by pedestal anti-resonant reflecting waveguide coupler,” IEEE Photon. Technol. Lett **12**, 1004–1006 (2000). [CrossRef]

2. B. E. Little, J. P. Laine, D. R. Lim, H. A. Haus, L. C. Kimerling, and S. T. Chu, “Pedestal antiresonant reflecting waveguides for robust coupling to microsphere resonators and for microphotonic circuits,” Opt. Lett **25**, 73–75 (2000). [CrossRef]

3. J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, “Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper,” Opt. Lett **22**, 1129–1131 (1997). [CrossRef] [PubMed]

4. J. P. Laine, C. Tapalian, B. Little, and H. Haus, “Acceleration sensor based on high-Q optical microsphere resonator and pedestal antiresonant reflecting waveguide coupler,” Sens. Actuators A**93**, 1–7 (2001). [CrossRef]

*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.

5. B. E. Little, J. P. Laine, and H. A. Haus, “Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators,” J. Lightwave Technol **17**, 704–715 (1999). [CrossRef]

7. A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett **36**, 321–322 (2000). [CrossRef]

## 2. Coupled resonator operation

### 2.1. Whispering gallery mode characteristics

5. B. E. Little, J. P. Laine, and H. A. Haus, “Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators,” J. Lightwave Technol **17**, 704–715 (1999). [CrossRef]

*α*is the evanescent field decay constant away from the sphere in the radial direction,

_{s}*β*is the modal propagation coefficient parallel to the surface of the sphere,

_{l}*R*is the physical sphere radius,

_{o}*n*is the refractive index of the sphere,

_{s}*n*is the refractive index of the medium surrounding the sphere, and λ is the free-space wavelength.

_{o}*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*, and each one corresponds to a different radial mode number,

_{l}*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.

*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

5. B. E. Little, J. P. Laine, and H. A. Haus, “Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators,” J. Lightwave Technol **17**, 704–715 (1999). [CrossRef]

^{2}is the proportion of power coupled from the waveguide mode to a specific WGM per revolution.

*E*is the WGM field of the sphere and

_{s}*E*(

_{w}*D*) is the waveguide evanescent field as a function of Teflon thickness in x-y cross-sectional plane.

_{teflon}*E*and

_{s}*E*(

_{w}*D*) are normalised such that ∬∣

_{teflon}*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*is the waveguide mode decay constant away from the waveguide surface.

_{wg}*K*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

_{o}6. A. W. Snyder and A. Ankiewicz, “Optical fiber couplers- optimum solution for unequal cores,” J. Lightwave Technol **6**, 463–474 (1988). [CrossRef]

^{-1}for

*l*=

*m*=83, δβ=0.92 μm

^{-1}for l =

*m*=77, the simple model gives accurate agreement with the experimental data.

*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

_{t}is a phase offset due to coupling to the waveguide.

*n*) and the coupling factor, κ, can be readily extracted from the experimental data for scattering using Eq. (9).

_{eff}R_{o}_{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}.

## 3. Coupled resonator measurement procedure

_{3}/KNO

_{3}/NaNO

_{3}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*∼ 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.

_{teflon}^{3+}-doped BK7 spheres of 15±1.5 μm in radius (Mo-sci) and, second, Nd

^{3+}-doped BK7 nominally 100 μm in radius fabricated in our laboratory. Neodymium is incorporated into the spheres to allow subsequent lasing studies [8

8. M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, “Fiber-coupled microsphere laser,” Opt. Lett **25**, 1430–1432 (2000). [CrossRef]

9. V. Sandoghdar, F. Treussart, J. Hare, V. Lefévre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev A **54**, R1777–R1780 (1996). [CrossRef]

10. G.Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, “Spectroscopic and lasing properties of Er^{3+}- doped glass microspheres,” J. Non-Cryst. Solids **352**, 2360–2363 (2006). [CrossRef]

## 4. Coupled resonator results and discussion

### 4.1. 15 μm radius sphere

*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

*δλ*is the width of the resonance lobe at full-width half-maximum power. The theoretical plot for the second-order (

_{FWHM}*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.

*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.

*R*) and refractive index (

_{o}*n*) 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

_{s}*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.

*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*=1.5004 and

_{s}*R*=15.387±0.007 μm. The full set of measured and theoretically predicted resonant wavelengths and assigned mode numbers,

_{o}*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.

*n*for each mode can be obtained by using Eq. (10) using the value of

_{eff}*FSR*found from the experimental data. The

*n*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 (

_{eff}*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

12. G. R. Elliott, D.W. Hewak, G. S. Murugan, and J. S. Wilkinson, “Chalcogenide glass microspheres; their production, characterization and potential,” Opt. Express **15**, 17542–17553 (2007). [CrossRef] [PubMed]

*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

*l*and

*n*assigned to each resonant wavelength, as before, by adjusting the radius,

*R*, 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 (

_{o}*n*=1,2, 3), for the best-fit values of

*R*=99.31±0.03 μm and

_{o}*n*=1.5006. The discrepancies between the theoretical and the experimental resonant wavelengths are less than 0.5 nm for all modes. Sphere mode numbers (

_{s}*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*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.

_{eff}## 5. Conclusions

*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.

## Acknowledgments

## References and links

1. | J. P. Laine, B. E. Little, D. R. Lim, H. C. Tapalian, L. C. Kimerling, and H. A. Haus, “Microsphere resonator mode characterization by pedestal anti-resonant reflecting waveguide coupler,” IEEE Photon. Technol. Lett |

2. | B. E. Little, J. P. Laine, D. R. Lim, H. A. Haus, L. C. Kimerling, and S. T. Chu, “Pedestal antiresonant reflecting waveguides for robust coupling to microsphere resonators and for microphotonic circuits,” Opt. Lett |

3. | J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, “Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper,” Opt. Lett |

4. | J. P. Laine, C. Tapalian, B. Little, and H. Haus, “Acceleration sensor based on high-Q optical microsphere resonator and pedestal antiresonant reflecting waveguide coupler,” Sens. Actuators A |

5. | B. E. Little, J. P. Laine, and H. A. Haus, “Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators,” J. Lightwave Technol |

6. | A. W. Snyder and A. Ankiewicz, “Optical fiber couplers- optimum solution for unequal cores,” J. Lightwave Technol |

7. | A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett |

8. | M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, “Fiber-coupled microsphere laser,” Opt. Lett |

9. | V. Sandoghdar, F. Treussart, J. Hare, V. Lefévre-Seguin, J. M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev A |

10. | G.Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, “Spectroscopic and lasing properties of Er |

11. | |

12. | G. R. Elliott, D.W. Hewak, G. S. Murugan, and J. S. Wilkinson, “Chalcogenide glass microspheres; their production, characterization and potential,” Opt. Express |

13. | A. E. Siegman |

**OCIS Codes**

(130.0130) Integrated optics : Integrated optics

(140.4780) Lasers and laser optics : Optical resonators

(230.5750) Optical devices : Resonators

(230.7390) Optical devices : Waveguides, planar

(130.2755) Integrated optics : Glass waveguides

(140.3945) Lasers and laser optics : Microcavities

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: May 23, 2008

Revised Manuscript: June 19, 2008

Manuscript Accepted: June 19, 2008

Published: July 9, 2008

**Citation**

Y. Panitchob, G. Senthil Murugan, M. N. Zervas, P. Horak, S. Berneschi, S. Pelli, G. Nunzi Conti, and J. S. Wilkinson, "Whispering gallery mode spectra of channel waveguide coupled Microspheres," Opt. Express **16**, 11066-11076 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-15-11066

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### References

- J. P. Laine, B. E. Little, D. R. Lim, H. C. Tapalian, L. C. Kimerling and H. A. Haus, "Microsphere resonator mode characterization by pedestal anti-resonant reflecting waveguide coupler," IEEE Photon. Technol. Lett. 12, 1004-1006 (2000). [CrossRef]
- B. E. Little, J. P. Laine, D. R. Lim, H. A. Haus, L. C. Kimerling and S. T. Chu, "Pedestal antiresonant reflecting waveguides for robust coupling to microsphere resonators and for microphotonic circuits," Opt. Lett. 25, 73-75 (2000). [CrossRef]
- J. C. Knight, G. Cheung, F. Jacques, and T. A. Birks, "Phase-matched excitation of whispering-gallery-mode resonances by a fiber taper," Opt. Lett. 22, 1129-1131 (1997). [CrossRef] [PubMed]
- J. P. Laine, C. Tapalian, B. Little, and H. Haus, "Acceleration sensor based on high-Q optical microsphere resonator and pedestal antiresonant reflecting waveguide coupler," Sens. Actuators A93, 1-7 (2001). [CrossRef]
- B. E. Little, J. P. Laine, and H. A. Haus, "Analytic theory of coupling from tapered fibers and half-blocks into microsphere resonators," J. Lightwave Technol. 17, 704-715 (1999). [CrossRef]
- A. W. Snyder and A. Ankiewicz, "Optical fiber couplers- optimum solution for unequal cores," J. Lightwave Technol. 6, 463-474 (1988). [CrossRef]
- A. Yariv, "Universal relations for coupling of optical power between microresonators and dielectric waveguides," Electron. Lett. 36, 321-322 (2000). [CrossRef]
- M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, "Fiber-coupled microsphere laser," Opt. Lett. 25, 1430-1432 (2000). [CrossRef]
- V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, "Very low threshold whispering-gallery-mode microsphere laser," Phys. Rev. A 54, R1777-R1780 (1996). [CrossRef]
- G. Nunzi Conti, A. Chiasera, L. Ghisa, S. Berneschi, M. Brenci, Y. Dumeige, S. Pelli, S. Sebastiani, P. Feron, M. Ferrari, and G. C. Righini, "Spectroscopic and lasing properties of Er3+- doped glass microspheres," J. Non-Cryst. Solids 352, 2360-2363 (2006). [CrossRef]
- http://www.Schott.com.
- G. R. Elliott, D.W. Hewak, G. S. Murugan, and J. S. Wilkinson, "Chalcogenide glass microspheres; their production, characterization and potential," Opt. Express 15, 17542-17553 (2007). [CrossRef] [PubMed]
- A. E. Siegman, Lasers (University Science Books, 1986).

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