## Photonic crystals as topological high-Q resonators |

Optics Express, Vol. 22, Issue 15, pp. 18579-18587 (2014)

http://dx.doi.org/10.1364/OE.22.018579

Acrobat PDF (1402 KB)

### Abstract

It is well known that defects, such as holes, inside an infinite photonic crystal can sustain localized resonant modes whose frequencies fall within a forbidden band. Here we prove that finite, defect-free photonic crystals behave as mirrorless resonant cavities for frequencies within but near the edges of an allowed band, regardless of the shape of their outer boundary. The resonant modes are extended, surface-avoiding (nearly-Dirichlet) states that may lie inside or outside the light cone. Independent of the dimensionality, quality factors and finesses are on the order of, respectively, *λ* is the vacuum wavelength and *L* >> *λ* is a typical size of the crystal. Similar topological modes exist in conventional Fabry-Pérot resonators, and in plasmonic media at frequencies just above those at which the refractive index vanishes.

© 2014 Optical Society of America

## 1. Introduction

*Q*, defined as the ratio between the frequency

*ω*and the width

*ω*is the separation between adjacent modes. The most common implementation of a RC involves a region defined by a mirrored surface. Mirrorless RCs are also well known. Examples include natural substances which rely on, e. g., total internal reflection to give confinement close to the boundary (as for whispering gallery modes [1

1. A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Sel. Top. Quantum Electron. **12**(1), 3–14 (2006). [CrossRef]

2. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. **58**(20), 2059–2062 (1987). [CrossRef] [PubMed]

3. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. **58**(23), 2486–2489 (1987). [CrossRef] [PubMed]

4. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**(6961), 944–947 (2003). [CrossRef] [PubMed]

5. R. Merlin, “Raman scattering by surface-avoiding acoustic phonons in semi-infinite superlattices,” Philos. Mag. B **70**(3), 761–766 (1994). [CrossRef]

6. M. Trigo, T. A. Eckhause, M. Reason, R. S. Goldman, and R. Merlin, “Observation of surface-avoiding waves: a new class of extended states in periodic media,” Phys. Rev. Lett. **97**(12), 124301 (2006). [CrossRef] [PubMed]

*Q*values on the order of

*λ*is the wavelength in vacuum and

7. M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B **65**(12), 125318 (2002). [CrossRef]

9. E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. **78**(2), 455–481 (2006). [CrossRef]

10. T. Xu, S. Yang, S. V. Nair, and H. E. Ruda, “Confined modes in finite-size photonic crystals,” Phys. Rev. B **72**(4), 045126 (2005). [CrossRef]

11. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. **40**(2), 453–488 (1961). [CrossRef]

*Q*modes discussed here should not be confused with the much-studied resonant guided modes supported by periodically patterned dielectric slabs [12

12. S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. **32**(14), 2606–2613 (1993). [CrossRef] [PubMed]

13. S. T. Thurman and G. M. Morris, “Controlling the spectral response in guided-mode resonance filter design,” Appl. Opt. **42**(16), 3225–3233 (2003). [CrossRef] [PubMed]

14. M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. **75**(3), 316–318 (1999). [CrossRef]

*λ*, which contain an infinite number of in-plane periods. For a lossless medium, their quality factors are thus dominated by out-of-plane radiative losses. In contrast, here we consider PC structures of arbitrary dimension, with a large but finite number of periods, and analyze the radiation from the edge of the periodic region.

## 2. Proof: two-dimensional photonic crystals

*E*modes in the two-dimensional case bar general arguments discussed later that apply as well to the vector wave equation in three dimensions. We assume that the PC is surrounded by vacuum and occupies a simply-connected region

*x*,

*y*) plane, of characteristic dimensions

*z*component of the electric field, Ψ, iswhere

*ε*is the permittivity, which is a periodic function of the position vector

**r =**(

*x*,

*y*) inside

*ρ*and

*ϕ*are the polar coordinates,

*s*along the curve. Explicit expressions of

*t*= 1, 2, ..) and

15. H. Feshbach, “On the Perturbation of Boundary Conditions,” Phys. Rev. **65**(11), 307–318 (1944). [CrossRef]

*G*in the Dirichlet set

*t*[15

15. H. Feshbach, “On the Perturbation of Boundary Conditions,” Phys. Rev. **65**(11), 307–318 (1944). [CrossRef]

15. H. Feshbach, “On the Perturbation of Boundary Conditions,” Phys. Rev. **65**(11), 307–318 (1944). [CrossRef]

*q*small compared to the size of the Brillouin zone, the states are approximately of the formwhere

*c*is the speed of light. Inserting the above expression in Eq. (8), we getwith, from Eq. (11),

*k*, the Green’s functions are

## 3. Examples

*T*for a periodic multilayer structure of total length

*D*and period

*d*. The frequency range shown is that of the second ‘optical’ band of the PC (in the first ‘acoustic’ band,

*ω*→ 0 for

*q*→ 0). The surface-avoiding, cavity modes occur near the edges of the allowed band, where the

*q*-dependence of

*ω*is quadratic. Figure 2(b) shows the intensity for the first three lowest-lying modes. Their envelopes are in a one-to-one correspondence with the intensity profiles of a mirrored RC (as well as with the quantum eigenfunctions of a particle in an infinitely deep potential well [17]), which can be ordered according to the number of zeros of the Dirichlet eigenfunctions. The results in Fig. 2(c) show that the distance in frequency between two arbitrary peaks and their width scale, respectively, like

*T*= 1 equals the number of cells [17].

18. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**(3), 173–190 (2001). [CrossRef] [PubMed]

*a*is the lattice parameter. As for the one-dimensional PC, the surface-avoiding modes manifest themselves as the narrow peaks that occur just below and above the edges of the allowed bands. Contour plots of the field magnitude are shown for the two highest-lying modes. Consistent with our theoretical description, the corresponding envelope functions show, respectively, no nodes and a line of nodes, in close correspondence with the associated Dirichlet modes. The simulations reveal similar behavior at the X-point for both the circle and the bow tie. Other boundaries also tested give the same outcome.

*R*are

*ω*-dependent refractive index. Therefore, for

*R*made of such a substance. Close to Re(

*ε*) = 0, and for

*A*is a constant and

20. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Annalen der Physik **330**(3), 377–445 (1908). [CrossRef]

*R*

^{−3}with increasing radius. The latter behavior persists until one reaches the point where the radiative width becomes smaller than the non-radiative one after which the peak intensity diminishes strongly. The existence of cavity-like, surface-avoiding modes can be established using simple, back-of-the-envelope arguments. The radiative lifetime of any given mode can be estimated aswhere

11. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. **40**(2), 453–488 (1961). [CrossRef]

*Q*~36000 have been reported [21

21. Q. Quan, I. B. Burgess, S. K. Y. Tang, D. L. Floyd, and M. Loncar, “High-Q, low index-contrast polymeric photonic crystal nanobeam cavities,” Opt. Express **19**(22), 22191–22197 (2011). [CrossRef] [PubMed]

22. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science **305**(5685), 847–848 (2004). [CrossRef] [PubMed]

*Q*~700 [23

23. K. Song and P. Mazumder, “Dynamic terahertz spoof surface plasmon-polariton switch based on resonance and absorption,” IEEE Trans. Electron. Dev. **58**(7), 2172–2176 (2011). [CrossRef]

6. M. Trigo, T. A. Eckhause, M. Reason, R. S. Goldman, and R. Merlin, “Observation of surface-avoiding waves: a new class of extended states in periodic media,” Phys. Rev. Lett. **97**(12), 124301 (2006). [CrossRef] [PubMed]

## Appendix

*λ*and that of the envelope function is

*p*>> 1 are exponentially small) and, thus,

## Acknowledgments

## References and links

1. | A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Sel. Top. Quantum Electron. |

2. | E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. |

3. | S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. |

4. | Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature |

5. | R. Merlin, “Raman scattering by surface-avoiding acoustic phonons in semi-infinite superlattices,” Philos. Mag. B |

6. | M. Trigo, T. A. Eckhause, M. Reason, R. S. Goldman, and R. Merlin, “Observation of surface-avoiding waves: a new class of extended states in periodic media,” Phys. Rev. Lett. |

7. | M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B |

8. | E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B |

9. | E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. |

10. | T. Xu, S. Yang, S. V. Nair, and H. E. Ruda, “Confined modes in finite-size photonic crystals,” Phys. Rev. B |

11. | A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. |

12. | S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. |

13. | S. T. Thurman and G. M. Morris, “Controlling the spectral response in guided-mode resonance filter design,” Appl. Opt. |

14. | M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. |

15. | H. Feshbach, “On the Perturbation of Boundary Conditions,” Phys. Rev. |

16. | E. M. Purcell, “Spontaneous Emission Probabilities at Radio Frequencies,” Phys. Rev. |

17. | For a discussion of the analogous problem of a particle in a finite one-dimensional periodic potential, see:D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys. |

18. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

19. | G. N. Watson, |

20. | G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Annalen der Physik |

21. | Q. Quan, I. B. Burgess, S. K. Y. Tang, D. L. Floyd, and M. Loncar, “High-Q, low index-contrast polymeric photonic crystal nanobeam cavities,” Opt. Express |

22. | J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science |

23. | K. Song and P. Mazumder, “Dynamic terahertz spoof surface plasmon-polariton switch based on resonance and absorption,” IEEE Trans. Electron. Dev. |

**OCIS Codes**

(230.5750) Optical devices : Resonators

(160.3918) Materials : Metamaterials

(160.5298) Materials : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: April 18, 2014

Revised Manuscript: July 9, 2014

Manuscript Accepted: July 14, 2014

Published: July 24, 2014

**Citation**

R. Merlin and S. M. Young, "Photonic crystals as topological high-Q resonators," Opt. Express **22**, 18579-18587 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-18579

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### References

- A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Sel. Top. Quantum Electron.12(1), 3–14 (2006). [CrossRef]
- E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett.58(20), 2059–2062 (1987). [CrossRef] [PubMed]
- S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett.58(23), 2486–2489 (1987). [CrossRef] [PubMed]
- Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature425(6961), 944–947 (2003). [CrossRef] [PubMed]
- R. Merlin, “Raman scattering by surface-avoiding acoustic phonons in semi-infinite superlattices,” Philos. Mag. B70(3), 761–766 (1994). [CrossRef]
- M. Trigo, T. A. Eckhause, M. Reason, R. S. Goldman, and R. Merlin, “Observation of surface-avoiding waves: a new class of extended states in periodic media,” Phys. Rev. Lett.97(12), 124301 (2006). [CrossRef] [PubMed]
- M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B65(12), 125318 (2002). [CrossRef]
- E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B71(19), 195122 (2005). [CrossRef]
- E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys.78(2), 455–481 (2006). [CrossRef]
- T. Xu, S. Yang, S. V. Nair, and H. E. Ruda, “Confined modes in finite-size photonic crystals,” Phys. Rev. B72(4), 045126 (2005). [CrossRef]
- A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J.40(2), 453–488 (1961). [CrossRef]
- S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt.32(14), 2606–2613 (1993). [CrossRef] [PubMed]
- S. T. Thurman and G. M. Morris, “Controlling the spectral response in guided-mode resonance filter design,” Appl. Opt.42(16), 3225–3233 (2003). [CrossRef] [PubMed]
- M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett.75(3), 316–318 (1999). [CrossRef]
- H. Feshbach, “On the Perturbation of Boundary Conditions,” Phys. Rev.65(11), 307–318 (1944). [CrossRef]
- E. M. Purcell, “Spontaneous Emission Probabilities at Radio Frequencies,” Phys. Rev.69(11), 681 (1946).
- For a discussion of the analogous problem of a particle in a finite one-dimensional periodic potential, see:D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys.61(12), 1118–1124 (1993).
- S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express8(3), 173–190 (2001). [CrossRef] [PubMed]
- G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (University Press, Cambridge, 1958), p. 198.
- G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Annalen der Physik330(3), 377–445 (1908). [CrossRef]
- Q. Quan, I. B. Burgess, S. K. Y. Tang, D. L. Floyd, and M. Loncar, “High-Q, low index-contrast polymeric photonic crystal nanobeam cavities,” Opt. Express19(22), 22191–22197 (2011). [CrossRef] [PubMed]
- J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science305(5685), 847–848 (2004). [CrossRef] [PubMed]
- K. Song and P. Mazumder, “Dynamic terahertz spoof surface plasmon-polariton switch based on resonance and absorption,” IEEE Trans. Electron. Dev.58(7), 2172–2176 (2011). [CrossRef]

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