## Tuning of whispering gallery modes of spherical resonators using an external electric field

Optics Express, Vol. 17, Issue 19, pp. 16465-16479 (2009)

http://dx.doi.org/10.1364/OE.17.016465

Acrobat PDF (423 KB)

### Abstract

In this paper we investigate the electrostriction effect on the whispering gallery modes (WGM) of polymeric microspheres and the feasibility of a WGM-based microsensor for electric field measurement. The electrostriction is the elastic deformation (strain) of a dielectric material under the force exerted by an electrostatic field. The deformation is accompanied by mechanical stress which perturbs the refractive index distribution in the sphere. Both strain and stress induce a shift in the WGM of the microsphere. In the present, we develop analytical expressions for the WGM shift due to electrostriction for solid and thin-walled hollow microspheres. Our analysis indicates that detection of electric fields as small as ~500V/m may be possible using water filled, hollow solid polydimethylsiloxane (PDMS) microspheres. The electric field sensitivities for solid spheres, on the other hand, are significantly smaller. Results of experiments carried out using solid PDMS spheres agree well with the analytical prediction.

© 2009 OSA

## 1. Introduction

*Q = λ/δλ*(

*λ*is the wavelength of the interrogating laser and

*δλ*is the linewidth of the observed mode). The proposed WGM applications include those in spectroscopy [1

1. W. von Klitzing, “Tunable whispering modes for spectroscopy and CQED Experiments,” New J. Phys. **3**, 14.1–14.14 (2001). [CrossRef]

2. M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, “Fiber-coupled microsphere laser,” Opt. Lett. **25**(19), 1430–1432 (2000). [CrossRef]

3. H. C. Tapalian, J. P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photon. Technol. Lett. **14**(8), 1118–1120 (2002). [CrossRef]

4. B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. **15**(6), 998–1005 (1997). [CrossRef]

5. B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyeler, and G. L. Bona,Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks,” IEEE J. Sel. Top. Quantum Electron. **5**(5), 1400–1406 (1999). [CrossRef]

6. V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Strain tunable high-*Q* optical microsphere resonator,” Opt. Commun. **145**(1-6), 86–90 (1998). [CrossRef]

3. H. C. Tapalian, J. P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photon. Technol. Lett. **14**(8), 1118–1120 (2002). [CrossRef]

7. F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. **80**(21), 4057–4059 (2002). [CrossRef]

8. S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett. **28**(4), 272–274 (2003). [CrossRef] [PubMed]

9. A. T. Rosenberger and J. P. Rezac, “Whispering-gallerymode evanescent-wave microsensor for trace-gas detection,” Proc. SPIE **4265**, 102–112 (2001). [CrossRef]

11. T. Ioppolo, M. Kozhevnikov, V. Stepaniuk, M. V. Otügen, and V. Sheverev, “Micro-optical force sensor concept based on whispering gallery mode resonators,” Appl. Opt. **47**(16), 3009–3014 (2008). [CrossRef] [PubMed]

12. T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, “High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres,” J. Appl. Phys. **105**(1), 013535 (2009). [CrossRef]

13. T. Ioppolo and M. V. Ötügen, ““Pressure Tuning of Whispering Gallery Mode Resonators,” J. Opt. Soc. Am. B **24**(10), 2721–2726 (2007). [CrossRef]

14. G. Guan, S. Arnold, and M. V. Ötügen, “Temperature Measurements Using a Micro-Optical Sensor Based on Whispering Gallery Modes,” AIAA J. **44**(10), 2385–2389 (2006). [CrossRef]

16. V. M. N. Passaro and F. De Leonardis, “Modeling and Design of a Novel High-Sensitivity Electric Field Silicon-on-Insulator Sensor Based on a Whispering-Gallery-Mode Resonator,” IEEE J. Sel. Top. Quantum Electron. **12**(1), 124–133 (2006). [CrossRef]

**c**ould also be used as fast, narrowband optical switches and filters.

## 2. Electrostatic Field-Induced Stress in a Solid Dielectric Sphere

*a*and inductive capacity

*ε*

_{1}_{,}embedded in an inviscid dielectric fluid of inductive capacity

*ε*. The sphere is subjected to a uniform electric field

_{2}*E*in the direction of negative

_{0}*z*as shown in Fig. 1 .

*a*and

_{1}*a*are defined by the Clausius-Mossotti law [15] leading to:for the fluid (medium 2). Here,

_{2}*ε*is the inductive capacity of vacuum, and

_{0}*k*(

*k = ε/ε*) is the dielectric constant. Using Eq. (5) and Eq. (12) the pressure acting at the dielectric interface is given by:where

_{0}*A’*and

*B’*are defined as: Equation (15) represents the pressure acting on the sphere surface due to the inductive capacity discontinuity at the sphere-fluid interface. Apart from this, the electric field induces a pressure perturbation in the fluid as well. This is given byFor gas media,

*k≈1*, thus

*δP*is negligible.

*A*and

_{n}*B*have to be evaluated. These coefficients are calculated by satisfying the following boundary conditionsThe coefficient

_{n}*A*and

_{n}*B*are determined by expanding the pressure

_{n}*P*in terms of Legendre series as follows:From Eq. (15), it can be noted that only two terms of the series in Eq. (20) are needed to describe the pressure distribution, from which the coefficients

*Z*are defined as:Plugging Eq. (8) and (11) and Eq. (20) and (21), into Eq. (19), the coefficients

_{n}*A*and

_{n}*B*are determinate as follows:The radial deformation can be determined by using Eq. (7):

_{n}## 3. WGM Shift in a Solid Sphere Due to Electrostriction

*r = a*and

*ϑ = π/2*) by plugging Eq. (22) into Eq. (23):

*da/a*, has a quadratic dependence on the electric field strength.

*dn*, in Eq. (2). Here we neglect the effect of the electric field on the index of refraction of the microsphere. The Neumann-Maxwell equations provide a relationship between stress and refractive index as follows [19

_{0}/n_{0}19. F. Ay, A. Kocabas, C. Kocabas, A. Aydinli, and S. Agan, “Prism coupling technique investigation of elasto-optical properties of thin polymer films,” J. Appl. Phys. **96**(12), 341–345 (2004). [CrossRef]

*C*and

_{1}*C*are the elasto-optical constants of the material. In our analysis we consider PDMS microspheres that are manufactured as described in Ref [15]. For PDMS these values are

_{2}*C*=

_{1}= C_{2}*C*= −1.75x10

^{−10}m

^{2}/N [20]. Thus, for a spherical sensor, the fractional change in the refractive index due to mechanical stress is reduced to:Thus, evaluating the appropriate expressions for stress in Eq. (8), 9, 10) at

*ϑ*=

*π/2*and

*r*=

*a*, and introducing them into Eq. (26) the relative change in the refractive index can be obtained. In order to evaluate the WGM shift due to the applied electric field, the constants

*a*and

_{1}*a*must be evaluated. Very few reliable measurements of these constants for solids have been reported in the literature. Unfortunately, to our knowledge there are no experimental measurements of

_{2}*a*and

_{1}*a*for polymeric material including PDMS. In our analysis we take the values developed for an ideal polar rubber [21

_{2}21. T. Yamwong, A. M. Voice, and G. R. Davies, “Electrostrictive response of an ideal polar rubber,” J. Appl. Phys. **91**(3), 1472–1476 (2002). [CrossRef]

*da/a*) and stress (

*dn*) effects on the WGM shifts due to an electric field are shown. The stress and strain have opposite effects on WGM shifts, but as seen in the figure, the strain effect dominates over that of stress and thus, the latter effect can be ignored in calculations. If we assume that the minimum measurable WGM shift is

_{0}/n_{0}*∆λ = λ/Q*, the measurement resolution is defined as

^{7}an electric field as small as ~20 kV/m can be resolved with a solid PDMS microsphere (polymeric base to curing agent ratio of 60:1 by volume).

## 4. Electrostatic Field-Induced Stress in a Hollow Dielectric Sphere

*ε*with inner radius

_{1}*a*and outer radius

*b*that is placed in a uniform dielectric fluid of inductive capacity

*ε*as shown in Fig. 3 . The shell is filled with a fluid of inductive capacity

_{2}*ε*As in the solid microsphere case, in order to determine the WGM shift, the strain distribution at the sphere outer surface must be known. In order to find this distribution the pressure acting at the surfaces, as well as the body force inside the shell has to be determined. In general, both the pressure and the body force are functions of the electric field distribution.

_{3}.*r,ϑ,ϕ*) is given as:where

*Ф*is the potential function. From the above equation, the potential function in each medium can be written as:Constant

*A*,

*B*,

*C*,

*D*are determined by satisfying the boundary condition at each interface, which are defined as:The coefficients are obtained by solving the following linear systemThe matrix coefficient

*α*and

_{ij}*γ*are presented in Appendix A. The electric field distribution in each medium is obtained byFrom the above equation each component of the electric field can be obtained, and are listed as follows:

_{i}*E*and

_{r}*E*are the radial and polar component of the electric field in each medium. As done for the solid sphere the surface force acting at each interface can be written aswhere

_{ϑ}*a*and

*b*represent the media on the two sides of the interface. Using Eq. (7) and Eq. (12) the pressure distributions at the inner and outer interface are given as follows:Where

*P*is the pressure at the inner surface of the shell, while

_{1,3}*P*is the pressure on the outer surface. The constant

_{1,2}*Z, Y, K*and

*W*are defined as: Note that these pressures are due to the inductive capacity discontinuity at the interface separating the media. If the hollow cavity is filled with a liquid (

*k>1*), there will be an increment of the fluid pressure due to electrostriction. This change in pressure due to applied electric field is given by [18]:The effect of the body force inside the shell due to the applied electrostatic field can be calculated using Eq. (4). Considering an isotropic dielectric, the first term on the right hand side of Eq. (4) becomes zero. However, the electric field within the shell is not constant, hence, the second term on the right hand side of Eq. (4) is finite. Using the expression given by Eq. (33), we can find the body force (per unit volume) as:where the constants

*B,*and

*C*are constants determined from Eq. (28), For a thin walled shell, the body force along the radial direction is nearly constant. In Fig. 4 , the net surface pressure distribution along the polar direction (ϑ) is compared to the distribution of radial and polar body force per unit volume times the shell thickness,

*Bt*.

*A*,

_{n}*B*,

_{n}*C*and

_{n}*D*are determined by satisfying the boundary conditions. The boundary conditions are defined as follow:The pressure acting at the boundaries of the hollow sphere can be expanded into Fourier-Legendre series asAgain, only two terms of the series are needed to represent the pressure on the inner and outer surfaces of the hollow sphere. These are:Substituting Eq. (46) into Eq. (45) and then into Eq. (44) we obtained the constants of Eq. (39). They are determined by solving the following two linear systems

_{n}*β*

_{ij,}ϕ_{i,}δ_{ij,}ρ_{i}_{j}are presented in Appendix A. Once the constants

*A*,

_{n}*B*,

_{n}*C*and

_{n}*D*are known, the change in WGM due to strain (

_{n}*da/a*) can be calculated by using Eq. (39). However, as discussed earlier,

*dn*, thus we neglect this effect on WGM shifts.

_{0}/n_{0}<<da/a*ϑ = π/2, r = b*) of a hollow PDMS microsphere of 600µm diameter and

*b/a*= 0.95 are shown in Fig. 5 . In this configuration, the PDMS shell is filled with and also surrounded by air (Note here that the stress effect is several orders of magnitude smaller than that of strain and hence, does not play a role in WGM shift). Comparing Fig. 5 to Fig. 2, we see that the effect of electric field on shape distortion of the spheres are opposite: The solid sphere becomes elongated in the direction of the static field. On the other hand, the hollow sphere elongates in the direction normal to the applied field.

*(ε*. For this, we consider the case of a thin spherical shell of PDMS that is filled with water

_{3}>ε_{2})*(k = 80.1)*and surrounded by air on the outside. Figure 6 illustrates the solution for this particular configuration. A comparison of Fig. 6 and 5 reveals that, filling the sphere with water increases the sensitivity significantly.

*Q-*factor of 10

^{7}, the resolution of the sensor is estimated to be ~500 V/m. The next question we address is: Can such a sensor be used to detect contaminants in surrounding medium? Figure 7 illustrates this. Using the same configuration as before (spherical PDMS shell filled with water inside and surrounded by air), the electric field applied on the sphere is kept at 10k V/m and the refractive index of the outside medium is changed. The resulting WGM shift is given in Fig. 7. Again, with

*Q*-factor of ~10

^{7}, the sensor can detect changes in the refractive index of ~10

^{−4}in a gas (at the wavelength λ = 1.312 μm) . Figure 7, when compared with the analysis of Ref [10], indicates a resolution improvement of at least an order of magnitude when the electric field is applied to the micro-sphere. These results shows that a sensor could be developed for the detection of contaminants both in air and in liquids.

## 5. Experiments

12. T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, “High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres,” J. Appl. Phys. **105**(1), 013535 (2009). [CrossRef]

11. T. Ioppolo, M. Kozhevnikov, V. Stepaniuk, M. V. Otügen, and V. Sheverev, “Micro-optical force sensor concept based on whispering gallery mode resonators,” Appl. Opt. **47**(16), 3009–3014 (2008). [CrossRef] [PubMed]

12. T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, “High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres,” J. Appl. Phys. **105**(1), 013535 (2009). [CrossRef]

^{6}during the experiments.

## 6. Conclusion

## Appendix

## Acknowledgments

## References and links

1. | W. von Klitzing, “Tunable whispering modes for spectroscopy and CQED Experiments,” New J. Phys. |

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

3. | H. C. Tapalian, J. P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photon. Technol. Lett. |

4. | B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. |

5. | B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyeler, and G. L. Bona,Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks,” IEEE J. Sel. Top. Quantum Electron. |

6. | V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Strain tunable high- |

7. | F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. |

8. | S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett. |

9. | A. T. Rosenberger and J. P. Rezac, “Whispering-gallerymode evanescent-wave microsensor for trace-gas detection,” Proc. SPIE |

10. | N. Das, T. Ioppolo, and V. Ötügen, “Investigation of a micro-optical concentration sensor concept based on whispering gallery mode resonators,” presented at the 45th AIAA Aerospace Sciences Meeting and Exhibition, Reno, Nev., January 8–11 2007. |

11. | T. Ioppolo, M. Kozhevnikov, V. Stepaniuk, M. V. Otügen, and V. Sheverev, “Micro-optical force sensor concept based on whispering gallery mode resonators,” Appl. Opt. |

12. | T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, “High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres,” J. Appl. Phys. |

13. | T. Ioppolo and M. V. Ötügen, ““Pressure Tuning of Whispering Gallery Mode Resonators,” J. Opt. Soc. Am. B |

14. | G. Guan, S. Arnold, and M. V. Ötügen, “Temperature Measurements Using a Micro-Optical Sensor Based on Whispering Gallery Modes,” AIAA J. |

15. | T. Ioppolo, U. K. Ayaz, M. V. Ötügen, and V. Sheverev, “A Micro-Optical Wall Shear Stress Sensor Concept Based on Whispering Gallery Mode Resonators,” 46th AIAA Aerospace Sciences Meeting and Exhibit, 8–11 January 2008. |

16. | V. M. N. Passaro and F. De Leonardis, “Modeling and Design of a Novel High-Sensitivity Electric Field Silicon-on-Insulator Sensor Based on a Whispering-Gallery-Mode Resonator,” IEEE J. Sel. Top. Quantum Electron. |

17. | R. W. Soutas-Little, |

18. | J. A. Stratton, |

19. | F. Ay, A. Kocabas, C. Kocabas, A. Aydinli, and S. Agan, “Prism coupling technique investigation of elasto-optical properties of thin polymer films,” J. Appl. Phys. |

20. | J. E. Mark, |

21. | T. Yamwong, A. M. Voice, and G. R. Davies, “Electrostrictive response of an ideal polar rubber,” J. Appl. Phys. |

22. | A. E. H. Love, |

23. | K. C. Kao, |

**OCIS Codes**

(040.0040) Detectors : Detectors

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(080.0080) Geometric optics : Geometric optics

(230.0230) Optical devices : Optical devices

**ToC Category:**

Detectors

**History**

Original Manuscript: June 1, 2009

Revised Manuscript: July 17, 2009

Manuscript Accepted: July 31, 2009

Published: September 1, 2009

**Citation**

Tindaro Ioppolo, Ulas Ayaz, and M. Volkan Ötügen, "Tuning of whispering gallery modes of spherical resonators using an external electric field," Opt. Express **17**, 16465-16479 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-19-16465

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### References

- W. von Klitzing, “Tunable whispering modes for spectroscopy and CQED Experiments,” New J. Phys. 3, 14.1–14.14 (2001). [CrossRef]
- M. Cai, O. Painter, K. J. Vahala, and P. C. Sercel, “Fiber-coupled microsphere laser,” Opt. Lett. 25(19), 1430–1432 (2000). [CrossRef]
- H. C. Tapalian, J. P. Laine, and P. A. Lane, “Thermooptical switches using coated microsphere resonators,” IEEE Photon. Technol. Lett. 14(8), 1118–1120 (2002). [CrossRef]
- B. E. Little, S. T. Chu, H. A. Haus, J. Foresi, and J.-P. Laine, “Microring resonator channel dropping filters,” J. Lightwave Technol. 15(6), 998–1005 (1997). [CrossRef]
- B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyeler, and G. L. Bona,Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks,” IEEE J. Sel. Top. Quantum Electron. 5(5), 1400–1406 (1999). [CrossRef]
- V. S. Ilchenko, P. S. Volikov, V. L. Velichansky, F. Treussart, V. Lefevre-Seguin, J.-M. Raimond, and S. Haroche, “Strain tunable high-Q optical microsphere resonator,” Opt. Commun. 145(1-6), 86–90 (1998). [CrossRef]
- F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80(21), 4057–4059 (2002). [CrossRef]
- S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett. 28(4), 272–274 (2003). [CrossRef] [PubMed]
- A. T. Rosenberger and J. P. Rezac, “Whispering-gallerymode evanescent-wave microsensor for trace-gas detection,” Proc. SPIE 4265, 102–112 (2001). [CrossRef]
- N. Das, T. Ioppolo, and V. Ötügen, “Investigation of a micro-optical concentration sensor concept based on whispering gallery mode resonators,” presented at the 45th AIAA Aerospace Sciences Meeting and Exhibition, Reno, Nev., January 8–11 2007.
- T. Ioppolo, M. Kozhevnikov, V. Stepaniuk, M. V. Otügen, and V. Sheverev, “Micro-optical force sensor concept based on whispering gallery mode resonators,” Appl. Opt. 47(16), 3009–3014 (2008). [CrossRef] [PubMed]
- T. Ioppolo, U. K. Ayaz, and M. V. Ötügen, “High-resolution force sensor based on morphology dependent optical resonances of polymeric spheres,” J. Appl. Phys. 105(1), 013535 (2009). [CrossRef]
- T. Ioppolo and M. V. Ötügen, “Pressure Tuning of Whispering Gallery Mode Resonators,” J. Opt. Soc. Am. B 24(10), 2721–2726 (2007). [CrossRef]
- G. Guan, S. Arnold, and M. V. Ötügen, “Temperature Measurements Using a Micro-Optical Sensor Based on Whispering Gallery Modes,” AIAA J. 44(10), 2385–2389 (2006). [CrossRef]
- T. Ioppolo, U. K. Ayaz, M. V. Ötügen, and V. Sheverev, “A Micro-Optical Wall Shear Stress Sensor Concept Based on Whispering Gallery Mode Resonators,” 46th AIAA Aerospace Sciences Meeting and Exhibit, 8–11 January 2008.
- V. M. N. Passaro and F. De Leonardis, “Modeling and Design of a Novel High-Sensitivity Electric Field Silicon-on-Insulator Sensor Based on a Whispering-Gallery-Mode Resonator,” IEEE J. Sel. Top. Quantum Electron. 12(1), 124–133 (2006). [CrossRef]
- R. W. Soutas-Little, Elasticity, (Dover Publications Inc., Mineola, NY, 1999).
- J. A. Stratton, Electromagnetic Theory (Mcgraw-Hill Book Company, Inc., New York and London, 1941).
- F. Ay, A. Kocabas, C. Kocabas, A. Aydinli, and S. Agan, “Prism coupling technique investigation of elasto-optical properties of thin polymer films,” J. Appl. Phys. 96(12), 341–345 (2004). [CrossRef]
- J. E. Mark, Polymer Data Handbook (Oxford University Press, 1999).
- T. Yamwong, A. M. Voice, and G. R. Davies, “Electrostrictive response of an ideal polar rubber,” J. Appl. Phys. 91(3), 1472–1476 (2002). [CrossRef]
- A. E. H. Love, The Mathematical Theory of Elasticity (Dover, 1926).
- K. C. Kao, Dielectric Phenomena in Solids (Elsevier Academic Press, 2004).

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