## Semiclassical evaluation of frequency splittings in coupled optical microdisks |

Optics Express, Vol. 21, Issue 20, pp. 24240-24253 (2013)

http://dx.doi.org/10.1364/OE.21.024240

Acrobat PDF (1129 KB)

### Abstract

When two microdisks are placed close to each other and the evanescent fields of their whispering gallery modes are overlapped, a strong coupling can be induced in the modes and lead to a doublet state. We attempt to evaluate the frequency splittings of the doublets by applying a semiclassical analysis in the regime of small wavelengths. Since a whispering gallery mode in a microdisk is a leaky mode, an established semiclassical method that deals with coupled closed systems is modified. As a result, we attain an analytic formula which can conveniently compute the frequency splittings of coupled whispering gallery modes. The derived formula is verified by demostrating a perfect agreement with numerical solutions of Maxwell’s equations.

© 2013 OSA

## 1. Introduction

2. M. Benyoucef, J.-B. Shim, J. Wiersig, and O. G. Schmidt, “Quality-factor enhancement of supermodes in coupled microdisks,” Opt. Lett. **36**, 1317–1319 (2011). [CrossRef] [PubMed]

3. M. Witzany, T.-L. Liu, J.-B. Shim, F. Hargart, E. Koroknay, W.-M. Schulz, M. Jetter, E. Hu, J. Wiersig, and P. Michler, “Strong mode coupling in InP quantum dot-based GaInP microdisk cavity dimers,” New J. Phys. **15**, 013060 (2013). [CrossRef]

4. S. Preu, H. G. L. Schwefel, S. Malzer, G. H. Döhler, L. J. Wang, M. Hanson, J. D. Zimmerman, and A. C. Gossard, “Coupled whispering gallery mode resonators in the terahertz frequency range.” Opt. Express **16**, 7336–7343 (2008). [CrossRef] [PubMed]

5. K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. **98**, 213904 (2007). [CrossRef] [PubMed]

6. U. Kuhl, S. Barkhofen, T. Tudorovskiy, H.-J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B **82**, 094308 (2010). [CrossRef]

7. M. Zhang, G. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. **109**, 233906 (2012). [CrossRef]

9. J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A **5**, 53–60 (2003). [CrossRef]

10. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. **181**, 687–702 (2010). [CrossRef]

11. S. V. Boriskina, “Spectrally engineered photonic molecules as optical sensors with enhanced sensitivity: a proposal and numerical analysis,” J. Opt. Soc. Am. B **23**, 1565–1573 (2006). [CrossRef]

*R*) can be derived as a function of the half distance between the centers of the two disks (

*d*), the resonant wave number (

*k*), and the azimuthal mode number (

*m*), Equation (2) is of course not as accurate as the computation with a basis of Bessel functions [11

11. S. V. Boriskina, “Spectrally engineered photonic molecules as optical sensors with enhanced sensitivity: a proposal and numerical analysis,” J. Opt. Soc. Am. B **23**, 1565–1573 (2006). [CrossRef]

*k*-vectors around coupled WGMs, since the equation is derived on the basis of ray dynamics.

## 2. Wilkinson’s formula

*V*and

_{L}*V*are the potential wells on the left-hand and the right-hand side, and

_{R}*ψ*and

_{L}*ψ*are the corresponding stationary states. In Ref. [12

_{R}12. C. Herring, “Critique of the Heitler-London method of calculating spin couplings at large distances,” Rev. Mod. Phys. **34**, 631–645 (1962). [CrossRef]

13. M. Wilkinson, “Tunnelling between tori in phase space,” Physica D **21**, 341–354 (1986). [CrossRef]

*I*in action-angle variables: [14], where

*S*is the generating function which has actions

**I**and coordinates

**r**as independent variables. Here, the generating function and the actions are defined in a classically forbidden region as well as in a classically allowed region.

*ħ*

^{2}) are neglected, because

*ħ*is supposed to be small enough.

*S/ħ*, and the integrand in Eq. (6) can be easily integrated by using Fresnel’s integral.

*x*

_{1}: and the probability amplitude along the combined trajectory is proportional to The above conditions are supposed to work also for tunneling. The phenomena of tunneling can be interpreted as a transfer of quanta following a trajectory in complex phase space in terms of the semiclassical physics. The system that we have is made up of two integrable systems, and the complex extensions of their manifolds intersect in the classically forbidden region. Then, tunneling trajectories can be determined on the combined complex manifold. Therefore, by taking the curve ∑ on the projection of the intersection onto configuration space and

*x*

_{1}as the coordinate on it, the trajectory which has the most contribution to tunneling can be identified and the integral in Eq. (6) can be evaluated in the form Eq. (10) is called Wilkinson’s formula. In Eq. (10)

*ω*and

_{L}*ω*are frequencies of vertical motions to ∑, and {

_{R}*I*,

_{R}*I*} is the Poisson bracket of the actions.

_{L}15. S. C. Creagh and M. D. Finn, “Evanescent coupling between discs: a model for near-integrable tunnelling,” J. Phys. A **34**, 3791–3801 (2001). [CrossRef]

## 3. Validity of semiclassical analysis for a single microdisk

*k*is excited in a microdisk with a homogeneous refractive index

*n*. Taking advantage of the analogy with quantum mechanics, the momentum of a ray can be set as

*nk*in the microdisk and as

*k*outside the microdisk by putting

*ħ*equal to 1. Then, the ray motion in the microdisk can be described by a particle dynamics in a shallow circular potential well, and its Hamiltonian is given by where The depth of this potential well is dependent upon

*k*

^{2}, and the total energy of the ray, (

*nk*)

^{2}/2 always lies above the well. However, such a shallow potential well can confine a ray motion inside of it, when its angular momentum is large enough to fulfill the inequality

*l*>

*kR*, which is exactly the same as the total internal reflection condition.

*l*as a constant of motion. Then, the radial motion is projected onto one dimensional oscillation with the effective Hamiltonian where

*l/nk*and

*R*in the effective potential well, and the region

*R*<

*r*<

*l/k*is classically forbidden. However, if a WGM corresponding to this ray motion is excited, this mode is able to leak out by tunneling through the effective potential barrier in the forbidden region.

*m*which is an integer, when the system is brought to wave mechanics. As the ray motion in the radial degree of freedom is oscillating, the trajectory forms an enclosed area in the corresponding phase space portrait. By setting the enclosed area in phase space equal to 2

*π*times an integer, we can derive a semiclassical quantization condition according to the WKB theory. In this derivation, the phase shifts associated with the both turning points in the effective potential well have to be taken into account. Because the reflecting boundary on the right-hand side of the confining one-dimensional potential in Fig. 2(a) is smooth, a constant phase shift of

*π*/2 is involved on this side. The phase shift on the other side is given by the Fresnel’s law as where

*ν*is a parameter, determined by the polarization of a mode. If the polarization is given such that its electric field is parallel to the cavity plane (Transverse Electric Mode),

*ν*is given by

*n*

^{2}. In the opposite case (Transverse Magnetic Mode), it is given by 1; [16]. Then, the quantization condition for a WGM can be formulated as where the integer

*N*is assigned as a radial mode number, and characterize a resonant mode together with a rotational mode number

*m*.

*S*

_{re}can be analytically derived in terms of a rescaled resonant wavenumber

*nkR*[14]. By substituting the analytic formula of

*S*

_{re}into Eq. (17), the following formula is obtained for the resonant wave number

*nkR*: Figure 3(a) shows the comparison between the result of Eq. (18) and the fulll solutions of Maxwell’s equations (see Appendix 5). It can be seen that the semiclassical calculation provides very good approximations of resonant wavenumbers.

*nkR*−

*i*Γ/(2

*nkR*)), we have verified that the semiclassical analysis is applicable to microdisks.

18. S.-Y. Lee and K. An, “Directional emission through dynamical tunneling in a deformed microcavity,” Phys. Rev. A **83**, 023827 (2011). [CrossRef]

*R*<

*r*<

*m/k*. Hereafter, we will call it the evanescent region.

## 4. Energy splittings in two coupled microdisks

15. S. C. Creagh and M. D. Finn, “Evanescent coupling between discs: a model for near-integrable tunnelling,” J. Phys. A **34**, 3791–3801 (2001). [CrossRef]

*ψ*(

**r**) is an analytic solution and

*ψ*

_{sc}(

**r**) is its semiclassical approximation. In the evanescent region (

*R*<

*r*<

*m/k*), a WGM is written in the analytic form where

*N*is a normalization factor. Conventionally, the norm of a WGM is defined by integrating the squared amplitude of the mode only in the interior of a microdisk: By means of Lommel’s integral [19], the integral in Eq. (25) can be evaluated as Since the complex extension of a WGM manifold exists only in the limited evanescent region as well as the evanescent coupling occurs in this region, the determination of the amplitude

_{m}20. R. Dubertrand, E. Bogomolny, N. Djellali, M. Lebental, and C. Schmit, “Circular dielectric cavity and its deformations,” Phys. Rev. A **77**, 013804 (2008). [CrossRef]

*ω*and

_{r}*k*are the frequency and the wavenumber of the radial ray motion. Then the semiclassically approximated wave function of a WGM is written as By comparing Eq. (30) to Eq. (32), the prefactor

_{r}^{2}must be multiplied to Wilkinson’s formula in Eq. (10).

*l*and

_{L}*l*as dynamical invariants, where

_{R}*d*is a half of the distance between two centers of microdisks. Thus the final form of the Wilkinson’s formula for coupled microdisks is written as For Eq. (35), the relationship is used. The result of Eq. (35) shows a good agreement with numerical calculation of coupled (21, 1)- and (23, 1)-WGMs, as can be seen in Fig. 6(a). For the numerical calculation, ‘Boundary Element Method’ is implemented [9

9. J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A **5**, 53–60 (2003). [CrossRef]

*m*,

*k*and

*d*. Then, by assigning an uncertainty-relation-like condition

*m*Δ

*θ*> 1, we derive the minimum distance

*d*for TM(

_{min}*m*, 1)-WGMs, until which Eq. (35) is valid. As can be intuitively predicted, Fig. 6(b) shows that

*d*gets smaller with increase of the angular mode number

_{min}*m*in the range of

*m*> 5. However, when

*m*gets less than 5, a drastic decrease of

*d*is noticed in Fig. 6(b). This aspect of

_{min}*d*cannot be viewed as a physical fact, because in this range the wavelength is comparable to the cavity size and the semiclassical approach is accordingly not valid. In a such range, coupled WGMs can show a discrepancy to a semiclassical approach [11

_{min}11. S. V. Boriskina, “Spectrally engineered photonic molecules as optical sensors with enhanced sensitivity: a proposal and numerical analysis,” J. Opt. Soc. Am. B **23**, 1565–1573 (2006). [CrossRef]

## 5. Conclusion and discussion

2. M. Benyoucef, J.-B. Shim, J. Wiersig, and O. G. Schmidt, “Quality-factor enhancement of supermodes in coupled microdisks,” Opt. Lett. **36**, 1317–1319 (2011). [CrossRef] [PubMed]

21. S. C. Creagh and M. White, “Differences between emission patterns and internal modes of optical resonators,” Phys. Rev. E **85**, 015201 (2012). [CrossRef]

22. J.-B. Shim, J. Wiersig, and H. Cao, “Whispering gallery modes formed by partial barriers in ultrasmall deformed microdisks,” Phys. Rev. E **84**, 035202 (2011). [CrossRef]

## Appendix

## Quasistationary modes of microdisks: solutions of Maxwell’s equations

*J*, because there is no source or sink in the cavity. Depending on the polarization of a mode, either internal electric field or magnetic field distribution of a mode can be set as

_{m}*ψ*(

**r**) =

*J*(

_{m}*nkr*)

*e*. At the boundary of the disk, this internal mode distribution and its normal derivative on the boundary should be equal to Hankel functions of the first kind

^{imϕ}*m*, the boundary matching equations are written as follows: where

*ν*is the same as in Eq. (16). Then the rescaled wavenumbers

*kR*stisfying the following condition, give us the resonances of a microdisk. As noticed in the main text, the resonant wavenumbers which fulfills the above equation are not given by real numbers, but by complex numbers with negative imaginary parts. If the value of

*m*is fixed, a radial mode number is assigned to every resonance in order of the magnitude of

*kR*.

## Asymptotic form of Hankel functions in evanescent region

*z*≫ 1):

## Acknowledgments

*Forschergruppe*FOR760.

## References and links

1. | K. Vahala, ed., |

2. | M. Benyoucef, J.-B. Shim, J. Wiersig, and O. G. Schmidt, “Quality-factor enhancement of supermodes in coupled microdisks,” Opt. Lett. |

3. | M. Witzany, T.-L. Liu, J.-B. Shim, F. Hargart, E. Koroknay, W.-M. Schulz, M. Jetter, E. Hu, J. Wiersig, and P. Michler, “Strong mode coupling in InP quantum dot-based GaInP microdisk cavity dimers,” New J. Phys. |

4. | S. Preu, H. G. L. Schwefel, S. Malzer, G. H. Döhler, L. J. Wang, M. Hanson, J. D. Zimmerman, and A. C. Gossard, “Coupled whispering gallery mode resonators in the terahertz frequency range.” Opt. Express |

5. | K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett. |

6. | U. Kuhl, S. Barkhofen, T. Tudorovskiy, H.-J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B |

7. | M. Zhang, G. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett. |

8. | J.-M. Jin, |

9. | J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A |

10. | A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. |

11. | S. V. Boriskina, “Spectrally engineered photonic molecules as optical sensors with enhanced sensitivity: a proposal and numerical analysis,” J. Opt. Soc. Am. B |

12. | C. Herring, “Critique of the Heitler-London method of calculating spin couplings at large distances,” Rev. Mod. Phys. |

13. | M. Wilkinson, “Tunnelling between tori in phase space,” Physica D |

14. | M. Brack and R. K. Bhaduri, |

15. | S. C. Creagh and M. D. Finn, “Evanescent coupling between discs: a model for near-integrable tunnelling,” J. Phys. A |

16. | M. Born and E. Wolf, |

17. | B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A |

18. | S.-Y. Lee and K. An, “Directional emission through dynamical tunneling in a deformed microcavity,” Phys. Rev. A |

19. | G. B. Arfken, |

20. | R. Dubertrand, E. Bogomolny, N. Djellali, M. Lebental, and C. Schmit, “Circular dielectric cavity and its deformations,” Phys. Rev. A |

21. | S. C. Creagh and M. White, “Differences between emission patterns and internal modes of optical resonators,” Phys. Rev. E |

22. | J.-B. Shim, J. Wiersig, and H. Cao, “Whispering gallery modes formed by partial barriers in ultrasmall deformed microdisks,” Phys. Rev. E |

23. | F. W. J. Oliver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(130.3120) Integrated optics : Integrated optics devices

(140.4780) Lasers and laser optics : Optical resonators

(140.5960) Lasers and laser optics : Semiconductor lasers

(080.1753) Geometric optics : Computation methods

(080.7343) Geometric optics : Wave dressing of rays

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: June 24, 2013

Revised Manuscript: August 16, 2013

Manuscript Accepted: August 28, 2013

Published: October 3, 2013

**Citation**

Jeong-Bo Shim and Jan Wiersig, "Semiclassical evaluation of frequency splittings in coupled optical microdisks," Opt. Express **21**, 24240-24253 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-20-24240

Sort: Year | Journal | Reset

### References

- K. Vahala, ed., Optical Microcavities(World Scientific, 2004).
- M. Benyoucef, J.-B. Shim, J. Wiersig, and O. G. Schmidt, “Quality-factor enhancement of supermodes in coupled microdisks,” Opt. Lett.36, 1317–1319 (2011). [CrossRef] [PubMed]
- M. Witzany, T.-L. Liu, J.-B. Shim, F. Hargart, E. Koroknay, W.-M. Schulz, M. Jetter, E. Hu, J. Wiersig, and P. Michler, “Strong mode coupling in InP quantum dot-based GaInP microdisk cavity dimers,” New J. Phys.15, 013060 (2013). [CrossRef]
- S. Preu, H. G. L. Schwefel, S. Malzer, G. H. Döhler, L. J. Wang, M. Hanson, J. D. Zimmerman, and A. C. Gossard, “Coupled whispering gallery mode resonators in the terahertz frequency range.” Opt. Express16, 7336–7343 (2008). [CrossRef] [PubMed]
- K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency,” Phys. Rev. Lett.98, 213904 (2007). [CrossRef] [PubMed]
- U. Kuhl, S. Barkhofen, T. Tudorovskiy, H.-J. Stöckmann, T. Hossain, L. de Forges de Parny, and F. Mortessagne, “Dirac point and edge states in a microwave realization of tight-binding graphene-like structures,” Phys. Rev. B82, 094308 (2010). [CrossRef]
- M. Zhang, G. Wiederhecker, S. Manipatruni, A. Barnard, P. McEuen, and M. Lipson, “Synchronization of micromechanical oscillators using light,” Phys. Rev. Lett.109, 233906 (2012). [CrossRef]
- J.-M. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley-IEEE, 2002).
- J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A5, 53–60 (2003). [CrossRef]
- A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. Joannopoulos, and S. G. Johnson, “Meep: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun.181, 687–702 (2010). [CrossRef]
- S. V. Boriskina, “Spectrally engineered photonic molecules as optical sensors with enhanced sensitivity: a proposal and numerical analysis,” J. Opt. Soc. Am. B23, 1565–1573 (2006). [CrossRef]
- C. Herring, “Critique of the Heitler-London method of calculating spin couplings at large distances,” Rev. Mod. Phys.34, 631–645 (1962). [CrossRef]
- M. Wilkinson, “Tunnelling between tori in phase space,” Physica D21, 341–354 (1986). [CrossRef]
- M. Brack and R. K. Bhaduri, Semiclassical Physics (Westview, 2008).
- S. C. Creagh and M. D. Finn, “Evanescent coupling between discs: a model for near-integrable tunnelling,” J. Phys. A34, 3791–3801 (2001). [CrossRef]
- M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
- B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A10, 343–352 (1993). [CrossRef]
- S.-Y. Lee and K. An, “Directional emission through dynamical tunneling in a deformed microcavity,” Phys. Rev. A83, 023827 (2011). [CrossRef]
- G. B. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).
- R. Dubertrand, E. Bogomolny, N. Djellali, M. Lebental, and C. Schmit, “Circular dielectric cavity and its deformations,” Phys. Rev. A77, 013804 (2008). [CrossRef]
- S. C. Creagh and M. White, “Differences between emission patterns and internal modes of optical resonators,” Phys. Rev. E85, 015201 (2012). [CrossRef]
- J.-B. Shim, J. Wiersig, and H. Cao, “Whispering gallery modes formed by partial barriers in ultrasmall deformed microdisks,” Phys. Rev. E84, 035202 (2011). [CrossRef]
- F. W. J. Oliver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions(Cambridge University, 2010).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.