## Controlled manipulation of mode splitting in an optical microcavity by two Rayleigh scatterers |

Optics Express, Vol. 18, Issue 23, pp. 23535-23543 (2010)

http://dx.doi.org/10.1364/OE.18.023535

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### Abstract

We report controlled manipulation of mode splitting in an optical microresonator coupled to two nanoprobes. It is demonstrated that, by controlling the positions of the nanoprobes, the split modes can be tuned simultaneously or individually and experience crossing or anti-crossing in frequency and linewidth. A tunable transition between standing wave mode and travelling wave mode is also observed. Underlying physics is discussed by developing a two-scatterer model which can be extended to multiple scatterers. Observed rich dynamics and tunability of split modes in a single microresonator will find immediate applications in optical sensing, opto-mechanics, filters and will provide a platform to study strong light-matter interactions in two-mode cavities.

© 2010 Optical Society of America

## 1. Introduction

1. K. J. Vahala, “Optical microcavities,” Nature **424**, 839–846 (2003). [CrossRef] [PubMed]

2. F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods **5**, 591–596 (2008). [CrossRef] [PubMed]

4. J. Zhu, Ş. K. Özdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics **4**, 46 (2009). [CrossRef]

5. T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. **95**, 033901 (2005). [CrossRef] [PubMed]

6. T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express **15**, 17172–17205 (2007). [CrossRef] [PubMed]

7. S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an Exceptional Point in a Chaotic Optical Microcavity,” Phys. Rev. Lett. **103**, 134101 (2009). [CrossRef] [PubMed]

8. T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, and K. J. Vahala, “Static Envelope Patterns in Composite Resonances Generated by Level Crossing in Optical Toroidal Microcavities,” Phys. Rev. Lett. **100**, 103905 (2008). [CrossRef] [PubMed]

12. L. Deych and J. Rubin, “Rayleigh scattering of whispering gallery modes of microspheres due to a single dipole scatterer,” Phys. Rev. A **80**, 061805 (2009). [CrossRef]

## 2. Experiments, theoretical model and discussions

*ω*

^{−}(lower frequency) and

*ω*

^{+}(higher frequency) modes with corresponding linewidths

*γ*

^{−}and

*γ*

^{+}(

*γ*

^{−}>

*γ*

^{+}), respectively. Then the second probe is introduced. This probe bends down and slides along the surface vertically as it contacts the rim of the microtoroid. Due to cone-like shape of the tip, vertical movement gradually increases its diameter allowing to simulate a scatterer of increasing size within mode volume without changing lateral position. This does not cause any significant damage to the microtoroid as witnessed by no observable change in the value of

*Q*factor. As the size increases, the nanoprobe starts disturbing the already established SWMs. The evolution of SWMs and the amount of disturbance applied to

*ω*

^{∓}and

*γ*

^{∓}depend on the size and location of the second probe relative to the first.

*ω*

^{+}mode experiences red shift and linewidth broadening with increasing size of the second probe, while the

*ω*

^{−}mode is not perturbed much. At a specific size both modes coincide, i.e.,

*ω*

^{−}=

*ω*

^{+}and

*γ*

^{−}=

*γ*

^{+}. Thus, a single resonance is seen in transmission spectrum. With further increase of the second scatterer’s size, the modes become separated with

*ω*

^{−}mode now having a larger linewidth than

*ω*

^{+}whose linewidth equals to the initial

*γ*

^{−}. This suggests that both frequency and linewidth crossings have occurred. At the crossing point, back-scattering into the resonator vanishes as the back-scattered fields from the two scatterers have the same strength but

*π*-phase shift. Thus the vanishing of backward reflection coupled to the taper suggests a transition from SWM to TWM. The conditions for these to take place will become clear in the discussion of the theoretical model below. This observation implies that SWM, which limits nonclassical features of coherent matter-cavity field interaction due to the position dependence of the coupling strength, can be eliminated using external tuning with nanoprobes. This is particularly crucial in ultra-high-

*Q*microresonators because SWMs are usually formed due to mode splitting caused by structural defects and material inhomogenety.

*ω*

^{+}mode with no significant disturbance to

*ω*

^{−}. Thus

*ω*

^{+}experiences linewidth broadening and red shift gradually approaching to

*ω*

^{−}. At a specific scatterer size, the frequency difference between the modes reduces from its initial value of 19.6 MHz to 7 MHz, and the linewidths become very close to each other. At this point, modes are strongly coupled to the scatterer and to each other. With further increase in size,

*ω*

^{−}strongly couples to the scatterer, and the modes start to repel each other leading to increased splitting. This suggests avoided-crossing of frequency and linewidth.

*ω*

^{−}and

*ω*

^{+}and induces frequency shift and linewidth broadening. The rate of change in

*ω*

^{−}is higher than that in

*ω*

^{+}suggesting that the scatterer has a greater overlap with

*ω*

^{−}.

11. A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled Coupling of Counterpropagating Whispering-Gallery Modes by a Single Rayleigh Scatterer: A Classical Problem in a Quantum Optical Light,” Phys. Rev. Lett. **99**, 173603 (2007). [CrossRef] [PubMed]

*α*

_{1}located at

**r**

_{1}in the resonator mode volume

*V*leads to a mode splitting quantified with the coupling strength 2

*g*

_{1}= −

*α*

_{1}

*f*

^{2}(

**r**

_{1})

*ω*

_{0}

*/V*and the additional linewidth broadening

*c*is the speed of light,

*ω*

_{0}is the resonance frequency before splitting, and

*f*

^{2}(

**r**

_{1}) is the spatial variation of the intensity of the initial WGM. The resulting two SWMs have periodic spatial distributions. A single scatterer locates itself at the anti-node (node) of

*ω*

^{−}(

*ω*

^{+}) with

*ϕ*= 0 (

*π*/2) where

*ϕ*denotes the spatial phase difference between the first scatterer and the anti-node of an SWM [4

4. J. Zhu, Ş. K. Özdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics **4**, 46 (2009). [CrossRef]

11. A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled Coupling of Counterpropagating Whispering-Gallery Modes by a Single Rayleigh Scatterer: A Classical Problem in a Quantum Optical Light,” Phys. Rev. Lett. **99**, 173603 (2007). [CrossRef] [PubMed]

*α*

_{2}is introduced at location

**r**

_{2}with a spatial phase difference of

*β*from the first scatterer, the already established SWMs redistribute themselves, and the amount of disturbance experienced by split modes depends on their overlap with the two scatterers [Fig. 3(a)]. Subsequently, the frequency shift (Δ

*ω*

^{−}=

*ω*

^{−}–

*ω*

_{0}) and the linewidth broadening (Δ

*γ*

^{−}=

*γ*

^{−}–

*γ*

_{0}) of

*ω*

^{−}with respect to the pre-scatterer resonance frequency

*ω*

_{0}and linewidth

*γ*

_{0}become where subscripts (1,2) represents the first and second scatterers. The cos

^{2}(·) terms scale the interaction strength depending on the position of the scatterer on SWMs [13

13. L. Chantada, N. I. Nikolaev, A. L. Ivanov, P. Borri, and W. Langbein, “Optical resonances in microcylinders: response to perturbations for biosensing,” J. Opt. Soc. Am. B **25**, 1312 (2008). [CrossRef]

*ω*

^{+}and Δ

*γ*

^{+}are obtained by replacing cos(·) with sin(·). Although we focus on two-scatterer case in this letter, this model can be extended to arbitrary number

*N*of scatterers by adding in Eqs. (1) and (2) the terms 2

*g*cos

_{i}^{2}(

*ϕ*–

*β*) and 2Γ

_{i}*cos*

_{i}^{2}(

*ϕ*–

*β*), respectively, for each of the 3 ≤

_{i}*i*≤

*N*scatterer. With the positions of the two scatterers fixed, the established orthonormal SWMs are distributed in such a way that the coupling rate between the two counter-propagating modes is maximized, in other words, the frequency splitting is maximized. It leads to: where

*ξ*=

*f*(

**r**

_{1})

*/f*(

**r**

_{2}) and

*χ*=

*α*

_{1}/

*α*

_{2}are positive real numbers. The two solutions of

*ϕ*have

*π*/2 phase difference and correspond to the two orthogonal SWMs. Verification of Eq. (3) is done by extensive finite-element simulations, and one example is presented in Fig. 3(b), where finite-element simulation results match very well with the calculated values from Eq. (3).

*δ*= Δ

*ω*

^{+}– Δ

*ω*=

^{−}*ω*

^{+}–

*ω*and

^{−}*ρ*= Δ

*γ*

^{−}– Δ

*γ*

^{+}as the frequency and linewidth differences of the resonance modes

*ω*

^{−}and

*ω*

^{+}, we find and from which frequency and linewidth crossings of the resonance modes can be calculated by setting

*δ*= 0 and

*ρ*= 0, respectively.

### A. Behavior of the frequencies of the resonance modes

*δ*= 0 which implies

*χ*

^{2}

*ξ*

^{4}+ 2

*χξ*

^{2}cos(2

*β*) + 1 = 0. It is satisfied only when cos(2

*β*) = −1 or

*β*=

*π*/2. In the following discussions we only consider 0 ≤

*β*≤

*π*/2, as cos(2

*β*) is an even function and has period of

*π*. Results of numerical simulations calculated from the model are shown in Fig. 4, which coincide very well with experimental observations in Fig. 2. Following are detailed discussions:

*β*= 0. We find*tan*(2*ϕ*) = 0, i.e.,*ϕ*= 0, implying that both particles locate at the antinode of*ω*^{−}, and Δ*γ*^{−}is maximized. This leads to*δ*= 2|*g*_{1}|(1 +*χ*^{−1}*ξ*^{−}^{2}). Thus decreasing*χ*increases*δ*by pushing*ω*^{−}further away from*ω*^{+}[Fig. 4(b)]. Note that if the size of the second scatterer reaches above Rayleigh limit [14], it may start disturbing14. R. G. Knollenberg, “The measurement of latex particle sizes using scattering ratios in the rayleigh scattering size range,” J. Aerosol Sci.

**20**, 331–345 (1989). [CrossRef]*ω*^{+}, too.*β*=*π*/2. The second particle stays at the anti-node of*ω*^{+}, thus increasing its size significantly affects the frequency and the linewidth of*ω*^{+}while its effect on*ω*^{−}is minimal. Then we find*δ*= 2|*g*_{1}|(1–*χ*^{−1}*ξ*^{−2}). This implies only one frequency crossing which occurs at*χ*=*ξ*^{−2}, i.e.,*g*_{1}=*g*_{2}. For*ξ*= 1, frequency crossing occurs at*χ*= 1 [Figs. 4(a)].- 0 <
*β*≤*π*/4. We have 0 ≤ cos2*β*< 1 (i.e., cosine is positive in the first quadrant of the unit circle) which implies that for a fixed*β*in this interval,*δ*is always greater than zero (*δ*> 0) and it increases with decreasing*χ*, that is with increasing*α*_{2}. Physically, this is understood as follows. The second scatterer affects both SWMs with strengths depending on its overlap with each mode. In this interval of*β*, the overlap of the second scatterer with*ω*^{−}mode is always larger than that with*ω*^{+}. Consequently, as*α*_{2}increases,*ω*^{−}mode is further red-shifted increasing*δ*. *π*/4 <*β*<*π*/2. We have −1 ≤ cos2*β*< 0 (i.e., cosine is negative in the second quadrant of the unit circle) implying that for a fixed*β*in this interval,*δ*is always greater than zero (*δ*> 0); however, contrary to the case (iii)*δ*has a minimum at*χ ξ*^{2}= 1/|cos(2*β*)| > 1 with*δ*_{min}= 2|*g*_{1}| (1 –*χ*^{−2}*ξ*^{−4})^{1/2}> 0. The physical process is explained as follows. In this case, too, the second scatterer affects both SWMs with strengths depending on its overlap with each mode. When the size of the second scatterer is small,*ω*^{+}feels it strongly and undergoes frequency shift coming closer to*ω*^{−}as*χ*decreases. This changes*ϕ*and increases the overlap of the second scatterer with*ω*^{−}leading to their stronger interaction which consequently, red-shifts*ω*^{−}and helps avoid crossing*ω*^{+}[Fig. 3(c)]. With a sufficiently large*α*_{2},*ϕ*>*π*/4 will be achieved which means*ω*^{−}will have larger overlap with the second scatterer than the first one.

### B. Behavior of the linewidths of the resonance modes

*ρ*= 0 in Eq. (5), we find the condition for linewidth crossing as 1 +

*ξ*

^{4}

*χ*

^{3}+

*χ*(1 +

*χ*)

*ξ*

^{2}cos(2

*β*) = 0 which can be satisfied only when cos(2

*β*) < 0 or

*π*/4 <

*β*≤

*π*/2, because both

*ξ*and

*χ*are positive real numbers.

*β*=*π*/2. We have*ϕ*= 0 and the two scatterers locate themselves at the anti-nodes of the each SWMs, i.e., 1st scatterer at*ω*^{−}and 2nd scatterer at*ω*^{+}. Thus scatterers independently affect the two SWMs. We find that a linewidth crossing takes place at*χ*^{2}*ξ*^{2}= 1, which means Γ_{1}= Γ_{2}. In this case if we also have*χ*=*ξ*= 1, which gives*g*_{1}=*g*_{2}and Γ_{1}= Γ_{2}implying the two SWMs have identical frequency and linewidth, but have orthogonal spatial distributions. The two SWMs merge to a TWM in the direction of the initial WGM. The other directional TWM vanishes as witnessed in experiments by vanishing of backward reflection in the fiber.*π*/4 <*β*<*π*/2. The roots of*ρ*= 0 can be found by setting*χ*^{3}*ξ*^{4}+*χ*(1 +*χ*)*ξ*^{2}cos(2*β*)+ 1 = 0. This is a transcendental third-order polynominal equation whose roots are too lengthy to give here. Given*π*/4 <*β*<*π*/2, either none or two positive real roots can be found for*χ*at specific values of*β*and*χ*. This suggests that two linewidth crossing points may be observed. Indeed the double crossing patterns are seen in the calculated patterns shown in Fig. 4(a) and 4(b). In both cases, one “symmetric” linewidth crossing [indicated by arrows in Fig. 4(a) and 4(b)] coincides with a frequency anti-crossing. From the plot of mode position*ϕ*, we see that the two SWMs “switch” distributions (*ϕ*shifts by*π/*2) around this point. This switching takes place around*χξ*^{2}= 1 and is the source of the symmetry of linewidth crossing. The other linewidth crossing takes place around*χ*^{2}*ξ*^{2}= 1, where the linewidth of one SWM changes significantly faster than that of the other one. This indicates that mode [red line in Fig. 4(a) and blue line in Fig. 4(b)] has much larger overlap with the second scatterer. Depending on whether*ξ*> 1 or*ξ*< 1, the “symmetric” linewidth crossing is observed before or after the other one. In experiments both scenarios were observed [Fig. 4(c) and 4(d)].Moreover, in the case that no positive real roots are found for*ρ*= 0, there is no linewidth crossing although one can always find*χ*for specific*ξ*and*β*which minimize*ρ*. On either side of this minimum,*ρ*increases implying linewidth anti-crossing. This can be explained in a similar way as the frequency anti-crossing when*π*/4 <*β*<*π*/2.- 0 ≤
*β*≤*π*/4. In this case*ρ*≠ 0 and similar to*δ*in this regime,*ρ*increases as*α*_{2}increases. Neither crossing nor anti-crossing can be observed.

7. S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an Exceptional Point in a Chaotic Optical Microcavity,” Phys. Rev. Lett. **103**, 134101 (2009). [CrossRef] [PubMed]

## 3. Conclusion

## Acknowledgments

## References and links

1. | K. J. Vahala, “Optical microcavities,” Nature |

2. | F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods |

3. | F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. U.S.A. |

4. | J. Zhu, Ş. K. Özdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics |

5. | T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, “Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity,” Phys. Rev. Lett. |

6. | T. J. Kippenberg and K. J. Vahala, “Cavity Opto-Mechanics,” Opt. Express |

7. | S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an Exceptional Point in a Chaotic Optical Microcavity,” Phys. Rev. Lett. |

8. | T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, and K. J. Vahala, “Static Envelope Patterns in Composite Resonances Generated by Level Crossing in Optical Toroidal Microcavities,” Phys. Rev. Lett. |

9. | D. S. Weiss, V. Sandoghdar, J. Hare, V. Lefvre-Seguin, J.-M. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced by light backscattering in silica microspheres,” Opt. Express |

10. | T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. |

11. | A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled Coupling of Counterpropagating Whispering-Gallery Modes by a Single Rayleigh Scatterer: A Classical Problem in a Quantum Optical Light,” Phys. Rev. Lett. |

12. | L. Deych and J. Rubin, “Rayleigh scattering of whispering gallery modes of microspheres due to a single dipole scatterer,” Phys. Rev. A |

13. | L. Chantada, N. I. Nikolaev, A. L. Ivanov, P. Borri, and W. Langbein, “Optical resonances in microcylinders: response to perturbations for biosensing,” J. Opt. Soc. Am. B |

14. | R. G. Knollenberg, “The measurement of latex particle sizes using scattering ratios in the rayleigh scattering size range,” J. Aerosol Sci. |

**OCIS Codes**

(140.4780) Lasers and laser optics : Optical resonators

(290.5870) Scattering : Scattering, Rayleigh

(140.3945) Lasers and laser optics : Microcavities

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: July 20, 2010

Revised Manuscript: September 22, 2010

Manuscript Accepted: October 16, 2010

Published: October 26, 2010

**Citation**

Jiangang Zhu, Şahin Kaya Özdemir, Lina He, and Lan Yang, "Controlled manipulation of mode splitting in an optical microcavity by two
Rayleigh scatterers," Opt. Express **18**, 23535-23543 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-23-23535

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### References

- K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003). [CrossRef] [PubMed]
- F. Vollmer, and S. Arnold, "Whispering-gallery-mode biosensing: label-free detection down to single molecules," Nat. Methods 5, 591-596 (2008). [CrossRef] [PubMed]
- F. Vollmer, S. Arnold, and D. Keng, "Single virus detection from the reactive shift of a whispering-gallery mode," Proc. Natl. Acad. Sci. U.S.A. 105, 20701-20704 (2008). [CrossRef] [PubMed]
- J. Zhu, S. K. Özdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, "On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator," Nat. Photonics 4, 46 (2009). [CrossRef]
- T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, "Analysis of Radiation-Pressure Induced Mechanical Oscillation of an Optical Microcavity," Phys. Rev. Lett. 95, 033901 (2005). [CrossRef] [PubMed]
- T. J. Kippenberg, and K. J. Vahala, "Cavity Opto-Mechanics," Opt. Express 15, 17172-17205 (2007). [CrossRef] [PubMed]
- S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, "Observation of an Exceptional Point in a Chaotic Optical Microcavity," Phys. Rev. Lett. 103, 134101 (2009). [CrossRef] [PubMed]
- T. Carmon, H. G. L. Schwefel, L. Yang, M. Oxborrow, A. D. Stone, and K. J. Vahala, "Static Envelope Patterns in Composite Resonances Generated by Level Crossing in Optical Toroidal Microcavities," Phys. Rev. Lett. 100, 103905 (2008). [CrossRef] [PubMed]
- D. S. Weiss, V. Sandoghdar, J. Hare, V. Lefvre-Seguin, J.-M. Raimond, and S. Haroche, "Splitting of high-Q Mie modes induced by light backscattering in silica microspheres," Opt. Express 20, 1835-1837 (1995).
- T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, "Modal coupling in traveling-wave resonators," Opt. Lett. 27, 1669 (2002). [CrossRef]
- A. Mazzei, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, "Controlled Coupling of Counterpropagating Whispering-Gallery Modes by a Single Rayleigh Scatterer: A Classical Problem in a Quantum Optical Light," Phys. Rev. Lett. 99, 173603 (2007). [CrossRef] [PubMed]
- L. Deych, and J. Rubin, "Rayleigh scattering of whispering gallery modes of microspheres due to a single dipole scatterer," Phys. Rev. A 80, 061805 (2009). [CrossRef]
- L. Chantada, N. I. Nikolaev, A. L. Ivanov, P. Borri, and W. Langbein, "Optical resonances in microcylinders: response to perturbations for biosensing," J. Opt. Soc. Am. B 25, 1312 (2008). [CrossRef]
- R. G. Knollenberg, "The measurement of latex particle sizes using scattering ratios in the Rayleigh scattering size range," J. Aerosol Sci. 20, 331-345 (1989). [CrossRef]

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